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%I #2 Mar 30 2012 17:34:33
%S 2,3,3,10,50,50,10,65,680,1775,1775,680,65,626,11542,53598,100554,
%T 100554,53598,11542,626,7777,229187,1745492,5202512,7950152,7950152,
%U 5202512,1745492,229187,7777,117650,5106474,59432614,274397694
%N A triangle sequence of polynomial coefficients: p(x,n)=(-1)^(n + 1)*(x - 1)^(3*n + 1)*Sum[(Binomial[m, n]* Binomial[m + 1, n + 1]/(m - n + 1))*(m + 1)^ n*x^m, {m, 0, Infinity}]/(x^n); t(n,m)=coefficients(p(x,n)+x^n*p(1/x,n))
%C Row sums are:
%C {2, 6, 120, 5040, 332640, 30270240, 3528645120, 502831929600, 84810985459200,
%C 16538142164544000, 3662446755711744000,...}
%F p(x,n)=(-1)^(n + 1)*(x - 1)^(3*n + 1)*Sum[(Binomial[m, n]* Binomial[m + 1, n + 1]/(m - n + 1))*(m + 1)^ n*x^m, {m, 0, Infinity}]/(x^n);
%F t(n,m)=coefficients(p(x,n)+x^n*p(1/x,n))
%e {2},
%e {3, 3},
%e {10, 50, 50, 10},
%e {65, 680, 1775, 1775, 680, 65},
%e {626, 11542, 53598, 100554, 100554, 53598, 11542, 626},
%e {7777, 229187, 1745492, 5202512, 7950152, 7950152, 5202512, 1745492, 229187, 7777},
%e {117650, 5106474, 59432614, 274397694, 612424944, 812843184, 812843184, 612424944, 274397694, 59432614, 5106474, 117650},
%e {2097153, 126054126, 2097410499, 14263234722, 47078954505, 85496057460, 102352156335, 102352156335, 85496057460, 47078954505, 14263234722, 2097410499, 126054126, 2097153},
%e {43046722, 3423843470, 77265218250, 732387384550, 3452145148910, 8922393746178, 13871595495670, 15346238845850, 15346238845850, 13871595495670, 8922393746178, 3452145148910, 732387384550, 77265218250, 3423843470, 43046722},
%e {1000000001, 101687151137, 2990117810108, 37744616613704, 242611866238742, 872095278964910, 1868330192416688, 2571940466451356, 2673255856625354, 2673255856625354, 2571940466451356, 1868330192416688, 872095278964910, 242611866238742, 37744616613704, 2990117810108, 101687151137, 1000000001},
%e {25937424602, 3282485943706, 121943246483490, 1978300682949474, 16662362139459228, 80125514615556636, 233339486568144348, 430592264194283100, 537393557602280652, 531006640383346764, 531006640383346764, 537393557602280652, 430592264194283100, 233339486568144348, 80125514615556636, 16662362139459228, 1978300682949474, 121943246483490, 3282485943706, 25937424602}
%t p[x_, n_] = (-1)^(n + 1)*(x - 1)^( 3*n + 1)*Sum[(Binomial[m, n]*Binomial[m + 1, n + 1]/(m - n + 1))*(m + 1)^n*x^m, {m, 0, Infinity}]/(x^n);
%t Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];
%t a = Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
%t Flatten[%]
%K nonn,tabl,uned
%O 0,1
%A _Roger L. Bagula_, Jan 26 2009
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