A155164 - OEIS

%I #5 Jan 27 2019 18:05:06

%S 1,2,6,6,34,52,34,315,525,525,315,3891,7956,6546,7956,3891,58828,

%T 153636,120176,120176,153636,58828,1048580,3405480,3219420,1950320,

%U 3219420,3405480,1048580,21523365,84108555,100695825,53131455,53131455

%N 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

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

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

%C 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.

%H Michael Z. Spivey and Laura L. Steil, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

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

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

%e {1},

%e {2},

%e {6, 6},

%e {34, 52, 34},

%e {315, 525, 525, 315},

%e {3891, 7956, 6546, 7956, 3891},

%e {58828, 153636, 120176, 120176, 153636, 58828},

%e {1048580, 3405480, 3219420, 1950320, 3219420, 3405480, 1048580},

%e {21523365, 84108555, 100695825, 53131455, 53131455, 100695825, 84108555, 21523365},

%e {500000005, 2289752440, 3390827500, 2109954760, 1062156190, 2109954760, 3390827500, 2289752440, 500000005},

%e {12968712306, 68202578598, 121311981780, 94003706412, 38734307304, 38734307304, 94003706412, 121311981780, 68202578598, 12968712306}

%t Clear[p, x, n, m];

%t 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}];

%t Table[(CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x] + Reverse[ CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]])/2, {n, 0, 10}];

%t Flatten[%]

%Y Cf. A001813.

%K nonn,tabl,uned

%O 0,2

%A _Roger L. Bagula_ and _Gary W. Adamson_, Jan 21 2009