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A triangle of polynomial coefficients: q(x,n)=-((x - 1)^(2*n + 1)/x^n)*Sum[(2*k + 1)^n*Binomial[k, n]*x^k, {k, 0, Infinity}]; p(x,n)=q(x,n)+x^n*q(1/x,n).
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%I #2 Mar 30 2012 17:34:33

%S 1,2,2,13,22,13,172,308,308,172,3281,7276,5766,7276,3281,80526,228822,

%T 174492,174492,228822,80526,2413405,8495474,8083699,4592764,8083699,

%U 8495474,2413405,85429688,359918120,440763192,220914920,220914920

%N A triangle of polynomial coefficients: q(x,n)=-((x - 1)^(2*n + 1)/x^n)*Sum[(2*k + 1)^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 {1, 4, 48, 960, 26880, 967680, 42577920, 2214051840, 132843110400,

%C 9033331507200, 686533194547200}

%F q(x,n)=-((x - 1)^(2*n + 1)/x^n)*Sum[(2*k + 1)^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 {1},

%e {2, 2},

%e {13, 22, 13},

%e {172, 308, 308, 172},

%e {3281, 7276, 5766, 7276, 3281},

%e {80526, 228822, 174492, 174492, 228822, 80526},

%e {2413405, 8495474, 8083699, 4592764, 8083699, 8495474, 2413405},

%e {85429688, 359918120, 440763192, 220914920, 220914920, 440763192, 359918120, 85429688},

%e {3487878721, 17132124952, 26131556188, 15925828264, 7488334150, 15925828264, 26131556188, 17132124952, 3487878721}, {161343848890, 905867202410, 1664943766280, 1285119074600, 499391861420, 499391861420, 1285119074600, 1664943766280, 905867202410, 161343848890},

%e {8339940489101, 52707061728718, 113751017120841, 108335058426024, 49780261735722, 20706515546388, 49780261735722, 108335058426024, 113751017120841, 52707061728718, 8339940489101}

%t Clear[p, x, n, m];

%t p[x_, n_] = -((x - 1)^(2*n + 1)/x^n)*Sum[( 2*k + 1)^n*Binomial[k, n]*x^k, {k, 0, Infinity}] Table[(FullSimplify[ExpandAll[p[x, n]]] + FullSimplify[ExpandAll[x^n*p[1/ x, n]]])/2, {n, 0, 10}];

%t Table[(CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x] + Reverse[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]])/2, {n, 0, 10}];

%t Flatten[%]

%Y A155164

%K nonn,tabl,uned

%O 0,2

%A _Roger L. Bagula_, Jan 30 2009