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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
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 (list; table; graph; refs; listen; history; text; internal format)
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 A228955 A226707

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

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Last modified June 19 23:01 EDT 2019. Contains 324222 sequences. (Running on oeis4.)