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%I #2 Mar 30 2012 17:34:40
%S 1,1,1,1,35,1,1,329,329,1,1,2535,6811,2535,1,1,18225,103925,103925,
%T 18225,1,1,127435,1384685,2868895,1384685,127435,1,1,881977,17115873,
%U 64568761,64568761,17115873,881977,1,1,6089807,202236439,1283008495
%N A symmetrical triangle of polynomial coefficients:q=4;p(x,n,q)=(1 - x)^(n + 1)*Sum[((q*k + 1)^n + (q*k + q - 1)^n)*x^k, {k, 0, Infinity}]
%C Row sums are:
%C {1, 2, 37, 660, 11883, 244302, 5893137, 165133224, 5284763991, 190253432610,
%C 7610144528061,...}.
%F q=4;p(x,n,q)=(1 - x)^(n + 1)*Sum[((q*k + 1)^n + (q*k + q - 1)^n)*x^k, {k, 0, Infinity}];
%F t(n,m,4)=coefficients(p(x,n,4));
%F Alternative polynomial function:
%F p(x,n,q)=q^n*(1 - x)^(1 + n)*(LerchPhi[x, -n, 1/q] + LerchPhi[x, -n, (-1 + q)/q])
%e {1},
%e {1, 1},
%e {1, 35, 1},
%e {1, 329, 329, 1},
%e {1, 2535, 6811, 2535, 1},
%e {1, 18225, 103925, 103925, 18225, 1},
%e {1, 127435, 1384685, 2868895, 1384685, 127435, 1},
%e {1, 881977, 17115873, 64568761, 64568761, 17115873, 881977, 1},
%e {1, 6089807, 202236439, 1283008495, 2302094507, 1283008495, 202236439, 6089807, 1},
%e {1, 42090209, 2323166957, 23495598125, 69265861013, 69265861013, 23495598125, 2323166957, 42090209, 1},
%e {1, 291532275, 26212748089, 406906029223, 1857593629387, 3028136650111, 1857593629387, 406906029223, 26212748089, 291532275, 1}
%t p[x_, n_, q_] = (1 - x)^(n + 1)* Sum[((q*k + 1)^n + (q*k + q - 1)^n)*x^k, {k, 0, Infinity}];
%t f[n_, m_, q_] := CoefficientList[FullSimplify[ExpandAll[p[x, n, q]]], x][[m + 1]];
%t Table[Flatten[Table[Table[FullSimplify[ ExpandAll[f[ n, m, q] - f[n, 0, q] + 1]], {m, 0, n}], {n, 0, 10}]], {q, 1, 10}]
%Y Cf. A008518, A174599
%K nonn,tabl,uned
%O 0,5
%A _Roger L. Bagula_, Apr 11 2010
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