%I #2 Mar 30 2012 17:34:33

%S 2,6,6,34,206,206,34,370,4840,14950,14950,4840,370,6642,142644,792216,

%T 1719618,1719618,792216,142644,6642,161294,5282074,45682504,158295424,

%U 274902544,274902544,158295424,45682504,5282074,161294,4827538

%N A triangle of polynomial coefficients: q(x,n)=(-1)^(n + 1)*(x - 1)^(3*n + 1)*Sum[(Binomial[m, n]* Binomial[m + 1, n + 1]/(m - n + 1))*(2*m + 1)^n*x^ m, {m, 0, Infinity}]/(x^n); p(x,n)=q(x,n)+x^n*q(1/x,n).

%C Row sums are:

%C {2, 12, 480, 40320, 5322240, 968647680, 225833287680, 64362486988800,

%C 21711612277555200, 8467528788246528000, 3750345477848825856000,...}.

%F q(x,n)=(-1)^(n + 1)*(x - 1)^(3*n + 1)*Sum[(Binomial[m, n]* Binomial[m + 1, n + 1]/(m - n + 1))*(2*m + 1)^n*x^ m, {m, 0, Infinity}]/(x^n);

%F p(x,n)=q(x,n)+x^n*q(1/x,n);

%F t(n,m)=coefficients(p(x,n))

%e {2},

%e {6, 6},

%e {34, 206, 206, 34},

%e {370, 4840, 14950, 14950, 4840, 370},

%e {6642, 142644, 792216, 1719618, 1719618, 792216, 142644, 6642},

%e {161294, 5282074, 45682504, 158295424, 274902544, 274902544, 158295424, 45682504, 5282074, 161294},

%e {4827538, 227651778, 2907137246, 14984780406, 38115062856, 56677184016, 56677184016, 38115062856, 14984780406, 2907137246, 227651778, 4827538},

%e {170861562, 11016050364, 197554369086, 1459983311028, 5313520312650, 10834039248120, 14364959341590, 14364959341590, 10834039248120, 5313520312650, 1459983311028, 197554369086, 11016050364, 170861562},

%e {6975764002, 589883814920, 14194396260000, 144086584363300, 732524911117760, 2067582128226648, 3567123431959120, 4329697827271850, 4329697827271850, 3567123431959120, 2067582128226648, 732524911117760, 144086584363300, 14194396260000, 589883814920, 6975764002},

%e {322687717462, 34650253894894, 1078225225888096, 14442860573483848, 98895486888500764, 381079250530358380, 884703838750948216, 1338789895455044032, 1514739863757428308, 1514739863757428308, 1338789895455044032, 884703838750948216, 381079250530358380, 98895486888500764, 14442860573483848, 1078225225888096, 34650253894894, 322687717462},

%e {16679881037250, 2217074072477334, 86648480181566430, 1481836294429602306, 13190542163684355876, 67274063665962266004, 208944364733797179732, 415175601761715156180, 565159740290092550148, 603857707780596736740, 603857707780596736740, 565159740290092550148, 415175601761715156180, 208944364733797179732, 67274063665962266004, 13190542163684355876, 1481836294429602306, 86648480181566430, 2217074072477334, 16679881037250}

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

%t p[x_, n_] = (-1)^(n + 1)*(x - 1)^(3*n + 1)*Sum[(Binomial[m, n]* Binomial[m + 1, n + 1]/(m - n + 1))*(2*m + 1)^n*x^ m, {m, 0, Infinity}]/(x^n);

%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 nonn,tabl,uned

%O 0,1

%A _Roger L. Bagula_, Jan 31 2009