convert a list into a series - Maple Help

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gfun[listtoseries] - convert a list into a series

gfun[seriestolist] - convert a series into a list

gfun[listtolist] - convert a list into a list

gfun[seriestoseries] - convert a series into a series

Calling Sequence

listtoseries(l, x, gf)

listtolist(l, gf)

seriestolist(s, gf)

seriestoseries(s, gf)

Parameters

l

-

list

x

-

name; unknown variable

gf

-

(optional) generating function type, e.g., 'egf' or 'ogf'

s

-

series

Description

• 

The listtoseries(l, x, gf) command accepts a list as input and returns a series.

  

By default, the listtoseries function creates a power series whose coefficients are exactly the elements of the list specified.

• 

The listtolist(l, gf) command accepts a list as input and returns a list.

  

By default, the listtolist function returns the input list unchanged.

• 

The seriestolist(s, gf) command accepts a series as input and returns a list.

  

By default, the seriestolist command returns a list whose entries are exactly the coefficients of the series specified.

• 

The seriestoseries(s, gf) command accepts a series as input and returns a series.

  

By default, the seriestoseries function returns the input series unchanged.

• 

Lists are viewed as lists of coefficients of power series and reciprocally series are viewed as generating series of lists of coefficients.

• 

If gf is specified, it is considered to be a type of generating function. The coefficients of the output are those of the corresponding generating function of the input.

  

E.g., listtoseries(l, x, 'egf') resp. listtolist(l, 'egf') returns the series resp. the list whose coefficients are li+1i! for 0i. Similarly, if s is a series with coefficients si for 0i, then seriestoseries(s, 'egf') resp. seriestolist(s, 'egf') returns the series resp. the list whose coefficients are sii!.

Examples

withgfun:

l:=1,1,2,5,14,42,132,429,1430,4862,16796,58786:

listtoseriesl,x,'egf'

1+x+x2+56x3+712x4+720x5+1160x6+1431680x7+1434032x8+2431181440x9+4199907200x10+41992851200x11+Ox12

(1)

The LambertW function is the inverse of yⅇy.  The following is a simple way to compute Taylor expansions of such functions.

S:=seriesyⅇy,y

S:=yy2+12y316y4+124y5+Oy6

(2)

seriestoseriesS,'revogf'

y+y2+32y3+83y4+12524y5+Oy6

(3)

You can then extract the list of coefficients.

L:=seriestolist

L:=0,1,1,32,83,12524

(4)

Multiplying the nth coefficient by n! recovers the number of planar trees of size n.

listtolistL,'Laplace'

0,1,2,9,64,625

(5)

See Also

gfun, gfun/gftypes, series, solve


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