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

Author

Roger L. Bagula, Jan 13 2009

Keywords

• nonn
• tabl
• easy

Comments

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

428554022400,...

Examples

			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;
		

Programs

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)