A154646 - OEIS

%I #6 Jul 23 2012 09:05:32

%S 2,3,3,5,26,5,9,153,153,9,17,796,2262,796,17,33,3951,25176,25176,3951,

%T 33,65,19266,243111,524876,243111,19266,65,129,93477,2168235,8760639,

%U 8760639,2168235,93477,129,257,453848,18445820,127880936,235517318

%N 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.

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

%C 428554022400,...

%e 2;

%e 3, 3;

%e 5, 26, 5;

%e 9, 153, 153, 9;

%e 17, 796, 2262, 796, 17;

%e 33, 3951, 25176, 25176, 3951, 33;

%e 65, 19266, 243111, 524876, 243111, 19266, 65;

%e 129, 93477, 2168235, 8760639, 8760639, 2168235, 93477, 129;

%e 257, 453848, 18445820, 127880936, 235517318, 127880936, 18445820, 453848, 257;

%p A154646 := proc(n,k)

%p     (-1)^(n+1)*(x-1)^(n+1)*add(x^j*((3*j+1)^n+(3*j+2)^n),j=0..k) ;

%p     coeftayl(%,x=0,k) ;

%p end proc: # _R. J. Mathar_, Jul 23 2012

%t Clear[p]; p[x_, n_] = (-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(3*m + 2)^n*x^m, {m, 0, Infinity}]

%t + (-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(3*m + 1)^n*x^m, {m, 0, Infinity}];

%t Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];

%t Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];

%t Flatten[%]

%t Contribution from _Roger L. Bagula_, Nov 27 2009: (Start)

%t p[t_] = Exp[t]*x/((-Exp[3*t] + x)) + Exp[2*t]*x/((-Exp[3*t] + x));

%t a = Table[ CoefficientList[FullSimplify[ExpandAll[(n!*(-1 + x)^(n + 1)/x)*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 10}];

%t Flatten[a] (End)

%K nonn,tabl,easy

%O 0,1

%A _Roger L. Bagula_, Jan 13 2009