A154646 Triangle T(n,k) with the coefficient [x^k] of the series (1-x)^(n+1)* sum_{m=0..infinity} [(3*m+1)^n + (3*m+2)^n]*x^m in row n, column k.

2, 3, 3, 5, 26, 5, 9, 153, 153, 9, 17, 796, 2262, 796, 17, 33, 3951, 25176, 25176, 3951, 33, 65, 19266, 243111, 524876, 243111, 19266, 65, 129, 93477, 2168235, 8760639, 8760639, 2168235, 93477, 129, 257, 453848, 18445820, 127880936, 235517318

Offset

0,1

Comments

Row sums are 2, 6, 36, 324, 3888, 58320, 1049760, 22044960, 529079040, 14285134080,

428554022400,...

Links

Table of n, a(n) for n=0..40.

Example

2;
3, 3;
5, 26, 5;
9, 153, 153, 9;
17, 796, 2262, 796, 17;
33, 3951, 25176, 25176, 3951, 33;
65, 19266, 243111, 524876, 243111, 19266, 65;
129, 93477, 2168235, 8760639, 8760639, 2168235, 93477, 129;
257, 453848, 18445820, 127880936, 235517318, 127880936, 18445820, 453848, 257;

Maple

A154646 := proc(n, k)
    (-1)^(n+1)*(x-1)^(n+1)*add(x^j*((3*j+1)^n+(3*j+2)^n), j=0..k) ;
    coeftayl(%, x=0, k) ;
end proc: # R. J. Mathar, Jul 23 2012

Mathematica

Clear[p]; p[x_, n_] = (-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(3*m + 2)^n*x^m, {m, 0, Infinity}]
+ (-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(3*m + 1)^n*x^m, {m, 0, Infinity}];
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
Flatten[%]
Contribution from Roger L. Bagula, Nov 27 2009: (Start)
p[t_] = Exp[t]*x/((-Exp[3*t] + x)) + Exp[2*t]*x/((-Exp[3*t] + x));
a = Table[ CoefficientList[FullSimplify[ExpandAll[(n!*(-1 + x)^(n + 1)/x)*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 10}];
Flatten[a] (End)

Crossrefs

Sequence in context: A270592 A096659 A154695 * A355868 A046826 A323713

Adjacent sequences: A154643 A154644 A154645 * A154647 A154648 A154649

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Jan 13 2009

Status

approved