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%I #2 Mar 30 2012 17:34:33
%S 2,4,4,48,48,728,232,232,728,20752,5312,1632,5312,20752,759132,168684,
%T 39864,39864,168684,759132,34016320,5788288,3904448,-2262272,3904448,
%U 5788288,34016320,1801010416,223429840,253864944,-64253360,-64253360
%N A triangle of polynomial coefficients: q(x,n)=-((x - 1)^(2*n + 1)/x^n)*Sum[(2*k + n)^n*Binomial[k, n]*x^k, {k, 0, Infinity}]; p(x,n)=q(x,n)+x^n*q(1/x,n).
%C Row sums are:
%C {2, 8, 96, 1920, 53760, 1935360, 85155840, 4428103680, 265686220800,
%C 18066663014400, 1373066389094400,...}.
%F q(x,n)=-((x - 1)^(2*n + 1)/x^n)*Sum[(2*k + n)^n*Binomial[k, n]*x^k, {k, 0, Infinity}];
%F p(x,n)=q(x,n)+x^n*q(1/x,n);
%F t(n,m)=coefficients(p(x,n))
%e {2},
%e {4, 4},
%e {48, 48},
%e {728, 232, 232, 728},
%e {20752, 5312, 1632, 5312, 20752},
%e {759132, 168684, 39864, 39864, 168684, 759132},
%e {34016320, 5788288, 3904448, -2262272, 3904448, 5788288, 34016320},
%e {1801010416, 223429840, 253864944, -64253360, -64253360, 253864944, 223429840, 1801010416},
%e {110076993792, 8135276544, 21010185216, -9977444352, 7196198400, -9977444352, 21010185216, 8135276544, 110076993792},
%e {7625557131380, 185854731220, 1792122898960, -827150318000, 256947063640, 256947063640, -827150318000, 1792122898960, 185854731220, 7625557131380},
%e {590491073741824, -15412908181504, 169164874601472, -90458315169792, 50709230659584, -35921522208768, 50709230659584, -90458315169792, 169164874601472, -15412908181504, 590491073741824}
%t Clear[p, x, n, m];
%t p[x_, n_] = -((x - 1)^(2*n + 1)/x^n)*Sum[(2*k + n)^n*Binomial[k, n]*x^k, {k, 0, Infinity}];
%t Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];
%t Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]
%t + Reverse[ CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]], {n, 0, 10}];
%t Flatten[%]
%K sign,tabl,uned
%O 0,1
%A _Roger L. Bagula_, Jan 31 2009
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