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 A155915 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). 0
 1, 2, 2, 13, 22, 13, 172, 308, 308, 172, 3281, 7276, 5766, 7276, 3281, 80526, 228822, 174492, 174492, 228822, 80526, 2413405, 8495474, 8083699, 4592764, 8083699, 8495474, 2413405, 85429688, 359918120, 440763192, 220914920, 220914920 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Row sums are: {1, 4, 48, 960, 26880, 967680, 42577920, 2214051840, 132843110400, 9033331507200, 686533194547200} LINKS FORMULA 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); t(n,m)=Coefficients(p(x,n)) EXAMPLE {1}, {2, 2}, {13, 22, 13}, {172, 308, 308, 172}, {3281, 7276, 5766, 7276, 3281}, {80526, 228822, 174492, 174492, 228822, 80526}, {2413405, 8495474, 8083699, 4592764, 8083699, 8495474, 2413405}, {85429688, 359918120, 440763192, 220914920, 220914920, 440763192, 359918120, 85429688}, {3487878721, 17132124952, 26131556188, 15925828264, 7488334150, 15925828264, 26131556188, 17132124952, 3487878721}, {161343848890, 905867202410, 1664943766280, 1285119074600, 499391861420, 499391861420, 1285119074600, 1664943766280, 905867202410, 161343848890}, {8339940489101, 52707061728718, 113751017120841, 108335058426024, 49780261735722, 20706515546388, 49780261735722, 108335058426024, 113751017120841, 52707061728718, 8339940489101} MATHEMATICA Clear[p, x, n, m]; 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}]; Table[(CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x] + Reverse[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]])/2, {n, 0, 10}]; Flatten[%] CROSSREFS Sequence in context: A074477 A141575 A151352 * A151367 A057648 A068511 Adjacent sequences:  A155912 A155913 A155914 * A155916 A155917 A155918 KEYWORD nonn,tabl,uned AUTHOR Roger L. Bagula, Jan 30 2009 STATUS approved

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