A171657 Coefficients of a set of infinite sum rational polynomials: p(x,n)=(-1 + x)^(m - 1)*( 1 - (1 + x)/(-1 + x))^(m + 1)*Sum[(k + 1)^(2*m - 1)*((x + 1)/( x - 1))^k, {k, 0, Infinity}]

1, 1, 0, -3, 4, 0, -30, 0, 30, 34, 0, -462, 0, 1050, 0, -630, 496, 0, -10560, 0, 40320, 0, -52920, 0, 22680, 11056, 0, -338448, 0, 1907400, 0, -4074840, 0, 3742200, 0, -1247400, 349504, 0, -14523600, 0, 112192080, 0, -344504160, 0, 505945440, 0

Offset

1,4

Comments

Row sums are:

{1, -2, 4, -8, 16, -32, 64, -128, 256, -512,...},

The infinite sums is found by doing a modular form substitution on the

Eulerian numbers A008292 infinite sum:

f((x+1)/(x-1))=(x-1)^(2*n)*f(x).

Further substitution of x^2->x removes the alternating zeros.

Links

Table of n, a(n) for n=1..46.

Formula

p(x,n)=(-1 + x)^(m - 1)*(1 - (1 + x)/(-1 + x))^(m + 1)*Sum[(k + 1)^(2*m - 1)*((x + 1)/(x - 1))^k, {k, 0, Infinity}]

Example

{1},
{1, 0, -3},
{4, 0, -30, 0, 30},
{34, 0, -462, 0, 1050, 0, -630},
{496, 0, -10560, 0, 40320, 0, -52920, 0, 22680},
{11056, 0, -338448, 0, 1907400, 0, -4074840, 0, 3742200, 0, -1247400},
{349504, 0, -14523600, 0, 112192080, 0, -344504160, 0, 505945440, 0, -356756400, 0, 97297200},
{14873104, 0, -804913392, 0, 8117600400, 0, -33403209840, 0, 69437768400, 0, -77416138800, 0, 44270226000, 0, -10216206000},
{819786496, 0, -55994442240, 0, 712037291520, 0, -3751484405760, 0, 10310701040640, 0, -16108264972800, 0, 14449801766400, 0, -6947020080000, 0, 1389404016000},
{56814228736, 0, -4778633088768, 0, 74633142796800, 0, -487475148142080, 0, 1695092146675200, 0, -3469875535526400, 0, 4330639993680000, 0, -3247037185392000, 0, 1346332491504000, 0, -237588086736000}

Mathematica

Clear[p, x, n]
p[x_, m_] = (-1 + x)^(m - 1)*(1 - (1 + x)/(-1 + x))^(m + 1)*Sum[(k + 1)^(2*m - 1)*((x + 1)/(x - 1))^k, {k, 0, Infinity}]
Table[CoefficientList[FullSimplify[ExpandAll[p[x, m]]], x], {m, 1, 10}]
Flatten[%]

Crossrefs

Cf. A008292

Sequence in context: A009126 A322278 A102222 * A287696 A366465 A084301

Adjacent sequences: A171654 A171655 A171656 * A171658 A171659 A171660

Keyword

sign,uned

Author

Roger L. Bagula, Dec 14 2009

Status

approved