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%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
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