A154594 A triangle sequence of polynomial coefficients:{a, b, c, d} = {3, 2, 2, 0}; p(x,n)=(-1)^(n)*(1 - d - c x)^(n + 1)*Sum[(a*k + b)^n*(c*x + d)^k, {k, 0, Infinity}].

1, -2, -2, 4, 26, 4, -8, -186, -240, -8, 16, 1090, 4524, 2008, 16, -32, -5866, -57992, -85424, -16288, -32, 64, 30354, 616452, 2099504, 1423968, 130848, 64, -128, -154202, -5902944, -39122296, -61925632, -22159968, -1048064, -128, 256

Offset

0,2

Comments

This result is from a scan of {a,b,c,d} that are quadratic symmetric.

Links

Table of n, a(n) for n=0..36.

Formula

{a, b, c, d} = {3, 2, 2, 0};

p(x,n)=(-1)^(n)*(1 - d - c x)^(n + 1)*Sum[(a*k + b)^n*(c*x + d)^k, {k, 0, Infinity}];

t(n,m)=coefficients(p(x,n)).

p(x,n)=(-2)^n *(1 - 2 x)^(1 + n)* LerchPhi[2 x, -n, 2/3]

Example

{1},
{-2, -2},
{4, 26, 4},
{-8, -186, -240, -8},
{16, 1090, 4524, 2008, 16},
{-32, -5866, -57992, -85424, -16288, -32},
{64, 30354, 616452, 2099504, 1423968, 130848, 64},
{-128, -154202, -5902944, -39122296, -61925632, -22159968, -1048064, -128},
{256, 776642, 53083228, 619239464, 1884138544, 1615232096, 331200832, 8387456, 256},
{-512, -3896010, -458838072, -8828796768, -46193602464, -76446547776, -38928658560, -4829723136, -67106304, -512},
{1024, 19508722, 3865505076, 117305639616, 982204711680, 2777786591040, 2766183413376, 889656803328, 69360887808, 536865280, 1024}

Mathematica

Clear[p, a, b, c, d, n];
{a, b, c, d} = {3, 2, 2, 0};
p[x_, n_] = (-1)^(n)*(1 - d - c x)^(n + 1)*Sum[(a*k + b)^n*(c*x + d)^k, {k, 0, Infinity}];
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
Flatten[%]

Crossrefs

Row sums are in A151919.

Sequence in context: A362766 A176161 A006829 * A098335 A346714 A049147

Adjacent sequences: A154591 A154592 A154593 * A154595 A154596 A154597

Keyword

sign,tabl,uned

Author

Roger L. Bagula, Jan 12 2009

Status

approved