A152300 A symmetrical triangle of coefficients of polynomials: q(x,n)=((1 - x)^(2*n)/(n*x))*Sum[Binomial[k + n - 1, k]*k^n*x^k, {k, 0, Infinity}]; p(x,n)=q(x,n)+x^(n-1)*q(1/x,n); t(n,m)=coefficients(p(x,n)).

2, 3, 3, 10, 20, 10, 65, 145, 145, 65, 626, 1612, 1572, 1612, 626, 7777, 24549, 23114, 23114, 24549, 7777, 117650, 450564, 496974, 340664, 496974, 450564, 117650, 2097153, 9493425, 12990807, 7851015, 7851015, 12990807, 9493425, 2097153

Offset

1,1

Comments

Row sums are: {2, 6, 40, 420, 6048, 110880, 2471040, 64864800, 1960358400, 67044257280,...}

Links

Table of n, a(n) for n=1..36.

Formula

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

p(x,n)=q(x,n)+x^(n-1)*q(1/x,n);

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

Example

{2},
{3, 3},
{10, 20, 10},
{65, 145, 145, 65},
{626, 1612, 1572, 1612, 626},
{7777, 24549, 23114, 23114, 24549, 7777},
{117650, 450564, 496974, 340664, 496974, 450564, 117650},
{2097153, 9493425, 12990807, 7851015, 7851015, 12990807, 9493425, 2097153},
{43046722, 225161564, 376201696, 262869988, 145798460, 262869988, 376201696, 225161564, 43046722},
{1000000001, 5937430213, 11798197840, 10137490792, 4649009794, 4649009794, 10137490792, 11798197840, 5937430213, 1000000001}

Mathematica

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

Crossrefs

Sequence in context: A094416 A218868 A329874 * A117030 A155758 A009097

Adjacent sequences: A152297 A152298 A152299 * A152301 A152302 A152303

Keyword

nonn

Author

Roger L. Bagula, Dec 02 2008

Status

approved