Symbolic sequences enter in various formulations in mathematics. This post is about a related new subpackage, Sequences, within the MathematicalFunctions package, available for download in Maplesoft's R&D page for Mathematical Functions and Differential Equations (currently bundled with updates to the Physics package). Perhaps the most typical cases of symbolic sequences are: 1) A sequence of numbers - say from n to m - frequently displayed as 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 2) A sequence of one object, say a, repeated say p times, frequently displayed as 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 3) A more general sequence, as in 1), but of different objects and not necessarily numbers, frequently displayed as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLzYlUSJuRidGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSgmY29tbWE7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYvNiVRJC4uLkYnRjJGNUY+LUYsNiVGLi1GLzYlUSJtRidGMkY1RjsvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnRkI= or likewise a sequence of functions 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 In all these cases, of course, none of n, m, or p are known: they are just symbols, or algebraic expressions, representing integer values. These most typical cases of symbolic sequences have been implemented in Maple since day 1 using the `$` operator. Cases 1), 2) and 3) above are respectively entered as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkYCRgRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiotRiw2JVEibkYnRi9GMi1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVC1GPjYtUSMuLkYnRkFGQ0ZGRkhGSkZMRk5GUC9GU1EsMC4yMjIyMjIyZW1GJ0ZVRj0tRiw2JVEibUYnRi9GMi8lK2V4ZWN1dGFibGVHRjEvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRkFGQUZpbkZbb0ZB, 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, and 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 or 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 To have computer algebra representations for all these symbolic sequences is something wonderful, I would say unique in Maple. Until recently, however, the typesetting of these symbolic sequences was frankly poor, input like 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 or `LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjJGBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUSJhRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJwRidGL0YyRkFGQUZB just being echoed in the display. More relevant: too little could be done with these objects; the rest of Maple didn't know how to add, multiply, differentiate or map an operation over the elements of the sequence, nor for instance count the sequence's number of elements. All this has now been implemented. What follows is a brief illustration. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRic= First of all, now these three types of sequences have textbook-like typesetting: LUkiJEclKnByb3RlY3RlZEc2IztJIm5HNiJJIm1HRig= LUkiJEclKnByb3RlY3RlZEc2IztJIm5HNiJJIm1HRig= LUkiJEclKnByb3RlY3RlZEc2JEkiYUc2IkkicEdGJw== LUkiJEclKnByb3RlY3RlZEc2JEkiYUc2IkkicEdGJw== For the above, a$p works the same way LUkiJEclKnByb3RlY3RlZEc2JCZJImFHNiI2I0kiaUdGKC9GKjtJIm5HRihJIm1HRig= LUkiJEclKnByb3RlY3RlZEc2JCZJImFHNiI2I0kiaUdGKC9GKjtJIm5HRihJIm1HRig= Moreover, this now permits textbook display of mathematical functions that depend on sequences of paramateters, for example: LUkqaHlwZXJnZW9tR0koX3N5c2xpYkc2IjYlNyMtSSIkRyUqcHJvdGVjdGVkRzYkJkkiYUdGJTYjSSJpR0YlL0YvOyIiIkkicEdGJTcjLUYpNiQmSSJiR0YlRi4vRi87RjJJInFHRiVJInpHRiU= LUkqaHlwZXJnZW9tR0koX3N5c2xpYkc2IjYlNyMtSSIkRyUqcHJvdGVjdGVkRzYkJkkiYUdGJTYjSSJpR0YlL0YvOyIiIkkicEdGJTcjLUYpNiQmSSJiR0YlRi4vRi87RjJJInFHRiVJInpHRiU= LUkwSW5jb21wbGV0ZUJlbGxCRzYiNiVJIm5HRiRJImtHRiQtSSIkRyUqcHJvdGVjdGVkRzYkLUkqZmFjdG9yaWFsR0YqNiNJImpHRiQvRi87IiIiLChGJkYyRichIiJGMkYy LUkwSW5jb21wbGV0ZUJlbGxCR0koX3N5c2xpYkc2IjYlSSJuR0YlSSJrR0YlLUkiJEclKnByb3RlY3RlZEc2JC1JKmZhY3RvcmlhbEdGKzYjSSJqR0YlL0YwOyIiIiwoRidGM0YoISIiRjNGMw== More interestingly, these new developments now permit differentiating these functions even when their arguments are symbolic sequences, and displaying the result as in textbooks, with copy and paste working properly, for instance LS9JJiVkaWZmRzYiSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkqaHlwZXJnZW9tR0koX3N5c2xpYkdGJTYlNyMtSSIkR0YnNiQmSSJhR0YlNiNJImlHRiUvRjQ7IiIiSSJwR0YlNyMtRi82JCZJImJHRiVGMy9GNDtGN0kicUdGJUkiekdGJUZB LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkqbWNvbXBsZXRlRzYkRiYvSSttb2R1bGVuYW1lR0YoRiQ2JC1JJW1yb3dHRi82JS1JJm1mcmFjR0YvNiQtSSNtb0dGLzYkUTAmRGlmZmVyZW50aWFsRDtGKC9JKm1hdGhjb2xvckdGKFEoIzkwOTA5MEYoLUY0NiRGOS1JI21pR0YvNiNRInpGKC1JJ21zcGFjZUdGLzYjL0kmd2lkdGhHRihRJjAuNGVtRigtRjQ2JS1JKW1zY3JpcHRzR0YvNiktRkM2JVEiRkYoL0kqZm9udHN0eWxlR0YoUSdub3JtYWxGKC9JK21zZW1hbnRpY3NHRi9RKmh5cGVyZ2VvbUYoLUkjbW5HRi82I1EicUYoLUklbm9uZUdGL0YoRmhuRmhuLUZlbjYjUSJwRihGaG4tRjo2I1EwJkFwcGx5RnVuY3Rpb247RigtSShtZmVuY2VkR0YvNiMtRjQ2Jy1GLjYkLUY0NictSSVtc3ViR0YvNiQtRkM2I1EiYUYoLUZlbjYjUSIxRigtRjo2I1EoJmNvbW1hO0YoLUZDNiNRJmAuLi5gRihGYnAtRmpvNiRGXHAtRkNGW28tRjQ2JS1Gam82JEZccC1GQzYjUSJpRigtRjo2I1EpJmRvbGxhcjtGKC1GNDYlRl9xLUY6NiNRKSZlcXVhbHM7RigtRjQ2JUZfcC1GOjYjUTEmcGVyaW9kOyZwZXJpb2Q7RihGanAtRjo2I1EnJnNlbWk7RigtRi42JC1GNDYnLUZqbzYkLUZDNiNRImJGKEZfcEZicEZlcEZicC1Gam82JEZoci1GQ0Zmbi1GNDYlLUZqbzYkRmhyRl9xRmJxLUY0NiVGX3FGZ3EtRjQ2JUZfcEZcckZdc0ZfckZCLUY0NiUtRkM2I1EmJWRpZmZGKEZdby1GYW82JEZMRkIqKC1JKHByb2R1Y3RHRiU2JCZJImFHRig2I0kiaUdGKC9GZHQ7IiIiSSJwR0YoRmd0LUZfdDYkJkkiYkdGKEZjdC9GZHQ7Rmd0SSJxR0YoISIiLUkqaHlwZXJnZW9tR0YnNiU3Iy1JIiRHRiY2JCwmRmF0Rmd0Rmd0Rmd0RmV0NyMtRmZ1NiQsJkZbdUZndEZndEZndEZddUkiekdGKEZndDcjLy1JJiVkaWZmR0YoNiQtRmJ1NiU3Iy1GZnVGYHQ3Iy1GZnVGanRGXXZGXXZGXXQ= It is very interesting how much this enhances the representation capabilities; to mention but one, this makes 100% possible the implementation of the Faa-di-Bruno formula for the nth symbolic derivative of composite functions (more on this in a post to follow this one). But the bread-and-butter first: the new package for handling sequences is LUkld2l0aEc2IjYjX0k2TWF0aGVtYXRpY2FsRnVuY3Rpb25zR0YkSSpTZXF1ZW5jZXNHRiQ= NydJJEFkZEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IkkuRGlmZmVyZW50aWF0ZUdGJ0kkTWFwR0YnSSlNdWx0aXBseUdGJEklTm9wc0dGJw== The five commands that got loaded do what their name tells. Consider for instance the first kind of sequences mentione above, i.e LUkiJEclKnByb3RlY3RlZEc2IztJIm5HNiJJIm1HRig= LUkiJEclKnByb3RlY3RlZEc2IztJIm5HNiJJIm1HRig= Check what is behind this nice typesetting LUknbHByaW50RyUqcHJvdGVjdGVkRzYjLUkiJEdGJDYjO0kibkc2IkkibUdGKw== `$`(n .. m) All OK. How many operands (an abstract version of Maple's nops command): LUklTm9wc0c2IjYjLUkiJEclKnByb3RlY3RlZEc2IztJIm5HRiRJIm1HRiQ= LChJIm1HNiIiIiJJIm5HRiQhIiJGJUYl That was easy, ok. Add the sequence LUkkQWRkRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMtSSIkR0YlNiM7SSJuR0YnSSJtR0Yn LCQqJiwoSSJtRzYiIiIiSSJuR0YmISIiRidGJ0YnLCZGKEYnRiVGJ0YnI0YnIiIj Multiply the sequence LUkpTXVsdGlwbHlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy1JIiRHRiU2IztJIm5HRidJIm1HRic= KiYtSSpmYWN0b3JpYWxHJSpwcm90ZWN0ZWRHNiNJIm1HNiIiIiItRiQ2IywmSSJuR0YoRikhIiJGKUYu Map an operation over the elements of the sequence LUkkTWFwRzYiNiRJImZHRiQtSSIkRyUqcHJvdGVjdGVkRzYjO0kibkdGJEkibUdGJA== LUkiJEclKnByb3RlY3RlZEc2JC1JImZHNiI2I0kiakc2Iy9JJ19sb2NhbEdGJEktX240NTczMDk5NjgwR0YoL0YqO0kibkdGKEkibUdGKA== LUknbHByaW50RyUqcHJvdGVjdGVkRzYjLUkiJEdGJDYkLUkiZkc2IjYjSSJqRzYjL0knX2xvY2FsR0YkSS1fbjQ1NzMwOTk2ODBHRisvRi07SSJuR0YrSSJtR0Yr `$`(f(j), j = n .. m) Map works as map, i.e. you can map extra arguments as well LUkkTWFwRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkkqU2VxdWVuY2VzRzYkRiUvRidJNk1hdGhlbWF0aWNhbEZ1bmN0aW9uc0dJKF9zeXNsaWJHRig2JUkkSW50R0YoLUkiJEdGJTYjO0kibkdGKEkibUdGKEkieEdGKA== LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSIkR0YmNiQtSSRJbnRHRiU2JEkiakc2Iy9JJ19sb2NhbEdGJkktX240NjAwMTQwMzg0R0YoSSJ4R0YoL0YyO0kibkdGKEkibUdGKEYr All this works the same way with symbolic sequences of forms 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 , and 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. For example: LUkiJEclKnByb3RlY3RlZEc2JEkiYUc2IkkicEdGJw== LUkiJEclKnByb3RlY3RlZEc2JEkiYUc2IkkicEdGJw== LUknbHByaW50RyUqcHJvdGVjdGVkRzYjLUkiJEdGJDYkSSJhRzYiSSJwR0Yq `$`(a, p) LUklTm9wc0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKlNlcXVlbmNlc0c2JEYlL0YnSTZNYXRoZW1hdGljYWxGdW5jdGlvbnNHSShfc3lzbGliR0YoNiMtSSIkR0YlNiRJImFHRihJInBHRig= SSJwRzYi LUkkQWRkRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMtSSIkR0YlNiRJImFHRidJInBHRic= KiZJImFHNiIiIiJJInBHRiRGJQ== LUkpTXVsdGlwbHlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy1JIiRHRiU2JEkiYUdGJ0kicEdGJw== KUkiYUc2IkkicEdGJA== Differentation also works LUkuRGlmZmVyZW50aWF0ZUc2IjYkLUkiJEclKnByb3RlY3RlZEc2JEkiYUdGJEkicEdGJEYq LUkiJEclKnByb3RlY3RlZEc2JCIiIkkicEc2Ig== LUkkTWFwRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkkqU2VxdWVuY2VzRzYkRiUvRidJNk1hdGhlbWF0aWNhbEZ1bmN0aW9uc0dJKF9zeXNsaWJHRig2JEkiZkdGKC1JIiRHRiU2JEkiYUdGKEkicEdGKA== LUkiJEclKnByb3RlY3RlZEc2JC1JImZHNiI2I0kiYUdGKEkicEdGKA== LUkuRGlmZmVyZW50aWF0ZUc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKlNlcXVlbmNlc0c2JEYlL0YnSTZNYXRoZW1hdGljYWxGdW5jdGlvbnNHSShfc3lzbGliR0YoNiQtSSIkR0YlNiQtSSJmR0YoNiNJImFHRihJInBHRihGNQ== LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSIkR0YmNiQtSSVkaWZmR0YmNiQtSSJmR0YoNiNJImFHRihGNUkicEdGKEYr For a symbolic sequence of type 3) LUkiJEclKnByb3RlY3RlZEc2JCZJImFHNiI2I0kiaUdGKC9GKjtJIm5HRihJIm1HRig= LUkiJEclKnByb3RlY3RlZEc2JCZJImFHNiI2I0kiaUdGKC9GKjtJIm5HRihJIm1HRig= LUklTm9wc0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKlNlcXVlbmNlc0c2JEYlL0YnSTZNYXRoZW1hdGljYWxGdW5jdGlvbnNHSShfc3lzbGliR0YoNiMtSSIkR0YlNiQmSSJhR0YoNiNJImlHRigvRjU7SSJuR0YoSSJtR0Yo LChJIm1HNiIiIiJJIm5HRiQhIiJGJUYl LUkkQWRkRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMtSSIkR0YlNiQmSSJhR0YnNiNJImlHRicvRi87SSJuR0YnSSJtR0Yn LUkkc3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQmSSJhR0YnNiNJImlHRicvRiw7SSJuR0YnSSJtR0Yn LUkpTXVsdGlwbHlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy1JIiRHRiU2JCZJImFHRic2I0kiaUdGJy9GLztJIm5HRidJIm1HRic= LUkocHJvZHVjdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkJkkiYUdGJzYjSSJpR0YnL0YsO0kibkdGJ0kibUdGJw== The following is nontrivial: differentiating the sequence 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, with respect to LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYlUSJrRidGL0YyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= should return 1 when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJrRidGNEY3Rj5GK0Y+ (i.e the running index has the value k), and 0 otherwise, and the same regarding m and k. That is how it works now: LUkuRGlmZmVyZW50aWF0ZUc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKlNlcXVlbmNlc0c2JEYlL0YnSTZNYXRoZW1hdGljYWxGdW5jdGlvbnNHSShfc3lzbGliR0YoNiQtSSIkR0YlNiQmSSJhR0YoNiNJImlHRigvRjU7SSJuR0YoSSJtR0YoJkYzNiNJImtHRig= LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSIkR0YmNiQtSSpQSUVDRVdJU0VHRiU2JDckIiIiL0kia0dGKEkiaUdGKDckIiIhSSpvdGhlcndpc2VHRigvRjY7SSJuR0YoSSJtR0YoNyMtRi02JC1JKnBpZWNld2lzZUdGJjYlRjRGM0Y4Rjo= LUknbHByaW50RyUqcHJvdGVjdGVkRzYjLUkiJEdGJDYkLUkqcGllY2V3aXNlR0YkNiUvSSJrRzYiSSJpR0YuIiIiIiIhL0YvO0kibkdGLkkibUdGLg== `$`(piecewise(k = i, 1, 0), i = n .. m) LUkkTWFwRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkkqU2VxdWVuY2VzRzYkRiUvRidJNk1hdGhlbWF0aWNhbEZ1bmN0aW9uc0dJKF9zeXNsaWJHRig2JEkiZkdGKC1JIiRHRiU2JCZJImFHRig2I0kiaUdGKC9GNjtJIm5HRihJIm1HRig= LUkiJEclKnByb3RlY3RlZEc2JC1JImZHNiI2IyZJImFHRig2I0kiaUdGKC9GLTtJIm5HRihJIm1HRig= LUkuRGlmZmVyZW50aWF0ZUc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJKlNlcXVlbmNlc0c2JEYlL0YnSTZNYXRoZW1hdGljYWxGdW5jdGlvbnNHSShfc3lzbGliR0YoNiQtSSIkR0YlNiQtSSJmR0YoNiMmSSJhR0YoNiNJImlHRigvRjg7SSJuR0YoSSJtR0YoJkY2NiNJImtHRig= LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSIkR0YmNiQqJi1JJWRpZmZHRiY2JC1JImZHRig2IyZJImFHRig2I0kiaUdGKEY2IiIiLUkqUElFQ0VXSVNFR0YlNiQ3JEY6L0kia0dGKEY5NyQiIiFJKm90aGVyd2lzZUdGKEY6L0Y5O0kibkdGKEkibUdGKDcjLUYtNiQqJkYwRjotSSpwaWVjZXdpc2VHRiY2JUY/RjpGQkY6RkQ= LUknbHByaW50RyUqcHJvdGVjdGVkRzYjLUkiJEdGJDYkKiYtSSVkaWZmR0YkNiQtSSJmRzYiNiMmSSJhR0YvNiNJImlHRi9GMSIiIi1JKnBpZWNld2lzZUdGJDYlL0kia0dGL0Y0RjUiIiFGNS9GNDtJIm5HRi9JIm1HRi8= `$`((diff(f(a[i]), a[i]))*piecewise(k = i, 1, 0), i = n .. m) And that is it. Summarizing: in addition to the former implementation of symbolic sequences, we now have textbook-like typesetting for them, and more important: Add, Multiply, Differentiate, Map and Nops. :) The first large application we have been working on taking advantage of this is symbolic differentiation, with very nice results; I will see to summarize them in a post to follow in a couple of days.