A155718 Symmetrical form of A039683 using polynomials: p(x,n)=Product[x - (2*i), {i, 0, Floor[n/2]}]/x; t(n,m)=coefficients(p(x,n)+x^n*p(1/x,n)); t(n,m)=A039683(n,m)+A039683(n,n-m).

2, -1, -1, 9, -12, 9, -47, 32, 32, -47, 385, -420, 280, -420, 385, -3839, 4354, -1460, -1460, 4354, -3839, 46081, -56490, 26684, -11760, 26684, -56490, 46081, -645119, 836296, -418936, 92624, 92624, -418936, 836296, -645119, 10321921, -14026824

Offset

0,1

Comments

Row sums are:

{2, -2, 6, -30, 210, -1890, 20790, -270270, 4054050, -68918850, 1309458150,...}.

The Stirling product form is: as even- odd factorization;

Product[x-i,{i,0,n}]=Product[x-(2*i),{i,0,Floor[n/2]}]*Product[x-(2*i+1),{i,0,Floor[n/2]}]

Links

Table of n, a(n) for n=0..37.

Formula

p(x,n)=Product[x - (2*i), {i, 0, Floor[n/2]}]/x;

t(n,m)=coefficients(p(x,n)+x^n*p(1/x,n));

t(n,m)=A039683(n,m)+A039683(n,n-m).

Example

{2},
{-1, -1},
{9, -12, 9},
{-47, 32, 32, -47},
{385, -420, 280, -420, 385},
{-3839, 4354, -1460, -1460, 4354, -3839},
{46081, -56490, 26684, -11760, 26684, -56490, 46081},
{-645119, 836296, -418936, 92624, 92624, -418936, 836296, -645119},
{10321921, -14026824, 7562120, -2189376, 718368, -2189376, 7562120, -14026824, 10321921},
{-185794559, 262803366, -150102120, 46239920, -7606032, -7606032, 46239920, -150102120, 262803366, -185794559},
{3715891201, -5441863790, 3264920736, -1076561200, 221207888, -57731520, 221207888, -1076561200, 3264920736, -5441863790, 3715891201}

Mathematica

Clear[p, x, n, b, a, b0];
p[x_, n_] := Product[x - (2*i), {i, 0, Floor[n/2]}]/x;
Table[Expand[ CoefficientList[ExpandAll[p[x, n]], x] + Reverse[CoefficientList[ExpandAll[p[x, n]], x]]], {n, 0, 20, 2}];
Flatten[%]

Crossrefs

A039683, A039757

Sequence in context: A119731 A368928 A283321 * A327088 A256168 A054768

Adjacent sequences: A155715 A155716 A155717 * A155719 A155720 A155721

Keyword

uned,sign

Author

Roger L. Bagula, Jan 25 2009

Status

approved