A155164 Polynomial triangle sequence of coefficients: p(x,n)=-((x - 1)^(2*n + 1)/x^n)*Sum[(k + 1)^n*Binomial[k, n]*x^k, {k, 0, Infinity}]. q(x,n)=(p(x,n)+x^n*p(1/x,n))/2

1, 2, 6, 6, 34, 52, 34, 315, 525, 525, 315, 3891, 7956, 6546, 7956, 3891, 58828, 153636, 120176, 120176, 153636, 58828, 1048580, 3405480, 3219420, 1950320, 3219420, 3405480, 1048580, 21523365, 84108555, 100695825, 53131455, 53131455

Offset

0,2

Comments

Row sums are:A001813; {1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400, 17643225600, 670442572800}.

This polynomial set is a new Binomial transform approach to infinite sums.

By the row sum it reaches the {2^n,(n+1)!,2^n*n!,(2*n+1)!!,(2*n)!/n!,...} fifth level of Sierpinski-Pascal complexity.

Links

Table of n, a(n) for n=0..33.

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Formula

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

q(x,n)=(p(x,n)+x^n*p(1/x,n))/2; t(n,m)=coefficients(q(x,n)).

Example

{1},
{2},
{6, 6},
{34, 52, 34},
{315, 525, 525, 315},
{3891, 7956, 6546, 7956, 3891},
{58828, 153636, 120176, 120176, 153636, 58828},
{1048580, 3405480, 3219420, 1950320, 3219420, 3405480, 1048580},
{21523365, 84108555, 100695825, 53131455, 53131455, 100695825, 84108555, 21523365},
{500000005, 2289752440, 3390827500, 2109954760, 1062156190, 2109954760, 3390827500, 2289752440, 500000005},
{12968712306, 68202578598, 121311981780, 94003706412, 38734307304, 38734307304, 94003706412, 121311981780, 68202578598, 12968712306}

Mathematica

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

Crossrefs

Cf. A001813.

Sequence in context: A069260 A056603 A019198 * A155948 A368393 A228955

Adjacent sequences: A155161 A155162 A155163 * A155165 A155166 A155167

Keyword

nonn,tabl,uned

Author

Roger L. Bagula and Gary W. Adamson, Jan 21 2009

Status

approved