Series[1/f ]

[Wouter Meeussen]

hi,

the series decomposition of 1/f[x] at x=0  is,
 with notation f[k] meaning the k-th derivative of f[x] at x=0 :

{
f[0]^(-1),
 -(f[1]/f[0]^2), 
(2*f[1]^2 - f[0]*f[2])/(2*f[0]^3),
 -(6*f[1]^3 - 6*f[0]*f[1]*f[2] + f[0]^2*f[3])/(6*f[0]^4),
 ...}

if f were such that f[0] = f[1] = ...=f[n] , as for the exponential function,
then this list reduces to (-1)^k 1/(f[0] k!)

we can however look at the numerators themselves by multiplying each term with
(f[0]^k k!) , and only then setting f[0]=f[1]=...f[n]

This produces polynomes in f[0] with coefficients that resist superseeker:

{{-1}, 
{2, -1}, 
{-6, 6, -1}, 
{24, -36, 14, -1}, 
{-120, 240, -150, 30, -1}, 
{720, -1800, 1560, -540, 62, -1}, 
{-5040, 15120, -16800, 8400, -1806, 126, -1}, 
{40320, -141120, 191520, -126000, 40824, -5796, 254, -1}, 
{-362880, 1451520, -2328480, 1905120, -834120, 186480, -18150, 510, -1}, 
{3628800, -16329600, 30240000, -29635200, 16435440, -5103000, 818520, -55980, 1022, -1}, 
{-39916800, 199584000, -419126400, 479001600, -322494480, 129230640, -29607600, 3498000, -171006, 2046, -1}, 
{479001600, -2634508800, 6187104000, -8083152000, 6411968640, -3162075840, 953029440, -165528000, 14676024, -519156, 4094, -1}}

the row-sums are {-1,1, ...}, the next to last element is 2^n-2, the first is (-1)^n n!
Anyone recognise this?
the absolute row-sums are A000670, so I checked the entry given there :
a(0) = 1, a(n) = Sum from k=1 to n of C(n,k)*a(n-k); 

and decomposed that into the following triangular table :
aa[0,0]=1;aa[n_,0]:=Sum[Binomial[n,k]*aa[n-k,0],{k,n}];
Table[aa[n,k]=Binomial[n,k]*aa[n-k,0],{n,12},{k,n}]


{{1}, 
{2, 1}, 
{9, 3, 1}, 
{52, 18, 4, 1}, 
{375, 130, 30, 5, 1}, 
{3246, 1125, 260, 45, 6, 1}, 
{32781, 11361, 2625, 455, 63, 7, 1}, 
{378344, 131124, 30296, 5250, 728, 84, 8, 1}, 
{4912515, 1702548, 393372, 68166, 9450, 1092,108, 9, 1}, 
{70872610, 24562575, 5675160, 983430, 136332, 15750, 1560, 135, 10, 1}, 
{1124723193, 389799355, 90062775, 15606690, 2163546, 249942, 24750, 2145, 165, 11, 1}, 
{19471590876, 6748339158, 1559197420, 270188325, 37456056, 4327092, 428472, 37125, 2860, 198, 12, 1}}

so the first triangular table is not a trivial re-hashing of the lower one (I think).





Wouter Meeussen
tel +32 (0)51 33 21 24
fax +32 (0)51 33 21 75
wouter.meeussen at vandemoortele.com