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%I #2 Mar 30 2012 17:34:32
%S 1,1,1,1,2,1,1,4,4,1,1,8,12,8,1,1,16,35,35,16,1,1,32,105,130,105,32,1,
%T 1,64,322,490,490,322,64,1,1,128,994,1967,2100,1967,994,128,1,1,256,
%U 3061,8232,9597,9597,8232,3061,256,1,1,512,9375,34855,48405,45654,48405
%N A triangular sequence of polynomial coefficients: p(x,n) = Sum[m^n*x^m/m!, {m, 0, Infinity}]/(x*Exp[x]); q(x,n)= If[n == 0, 1, p(x, n) + x^n*p(1/x, n)].
%C Row sums are:
%C {1, 2, 4, 10, 30, 104, 406, 1754, 8280, 42294, 231950,...}
%F p(x,n) = Sum[m^n*x^m/m!, {m, 0, Infinity}]/(x*Exp[x]);
%F q(x,n)= If[n == 0, 1, p(x, n) + x^n*p(1/x, n)];
%F t(n,m)=coefficients(q(x,n)).
%e {1},
%e {1, 1},
%e {1, 2, 1},
%e {1, 4, 4, 1},
%e {1, 8, 12, 8, 1},
%e {1, 16, 35, 35, 16, 1},
%e {1, 32, 105, 130, 105, 32, 1},
%e {1, 64, 322, 490, 490, 322, 64, 1},
%e {1, 128, 994, 1967, 2100, 1967, 994, 128, 1},
%e {1, 256, 3061, 8232, 9597, 9597, 8232, 3061, 256, 1},
%e {1, 512, 9375, 34855, 48405, 45654, 48405, 34855, 9375, 512, 1}
%t Clear[p]; p[x_, n_] = Sum[m^n*x^m/m!, {m, 0, Infinity}]/(x*Exp[x]);
%t q[x_, n_] = If[n == 0, 1, p[x, n] + x^n*p[1/x, n]];
%t Table[FullSimplify[ExpandAll[q[x, n]]], {n, 0, 10}];
%t Table[CoefficientList[FullSimplify[ExpandAll[q[x, n]]], x], {n, 0, 10}];
%t Flatten[%]
%K nonn,uned,tabl
%O 0,5
%A _Roger L. Bagula_, Jan 16 2009
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