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A137862 Triangular sequence of coefficients of the expansion of a degenerate partition of Chebyshev U(x,n);A053117 and Hermite H(x,n);A060821 functions: 1) f(x,t)=1/(1-2*x*t+t^2); 2) g(x,t)=Exp[2*x*t-t^2]; to give: p(x,t)=Exp[2*x*t-t^2]/(1-2*x*t+t^2).

%I A137862 #3 Mar 30 2012 17:34:26
%S A137862 1,0,4,-4,0,20,0,-60,0,128,60,0,-768,0,1040,0,1920,0,-10400,0,10432,
%T A137862 -1920,0,46800,0,-156480,0,125248,0,-109200,0,1095360,0,-2630208,0,
%U A137862 1753600,109200,0,-4381440,0,26302080,0,-49100800,0,28057856,0,9858240,0,-157812480,0,662860800,0,-1010082816,0
%N A137862 Triangular sequence of coefficients of the expansion of a degenerate partition of Chebyshev U(x,n);A053117 and Hermite H(x,n);A060821 functions: 1) f(x,t)=1/(1-2*x*t+t^2); 2) g(x,t)=Exp[2*x*t-t^2]; to give: p(x,t)=Exp[2*x*t-t^2]/(1-2*x*t+t^2).
%C A137862 Row sums are:
%C A137862 {1, 4, 16, 68, 332, 1952, 13648, 109552, 986896, 9865664, 108500864};
%F A137862 p(x,t)=Exp[2*x*t-t^2]/(1-2*x*t+t^2)=Sum(P(x,n)*t^n/n!,{n,0,Infinity}); out_n,m=n!*Coefficients(P(x,n)).
%e A137862 {1},
%e A137862 {0, 4},
%e A137862 {-4, 0, 20},
%e A137862 {0, -60, 0, 128},
%e A137862 {60, 0, -768, 0,1040},
%e A137862 {0, 1920, 0, -10400, 0, 10432},
%e A137862 {-1920, 0, 46800, 0, -156480, 0, 125248},
%e A137862 {0, -109200, 0, 1095360, 0, -2630208, 0, 1753600},
%e A137862 {109200, 0, -4381440, 0, 26302080, 0, -49100800, 0, 28057856},
%e A137862 {0, 9858240, 0, -157812480, 0, 662860800, 0, -1010082816, 0, 505041920}, {-9858240, 0, 591796800, 0, -5523840000, 0, 17676449280, 0, -22726886400, 0, 10100839424}
%t A137862 Clear[p, b, a]; p[t_] = FullSimplify[(1/(1 - 2*x*t + t^2))*Exp[2*x*t - t^2]]; Table[ ExpandAll[n!*SeriesCoefficient[Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[n!*SeriesCoefficient[Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]
%Y A137862 Cf. A060821, A053117.
%K A137862 tabl,uned,sign
%O A137862 1,3
%A A137862 _Roger L. Bagula_, Apr 29 2008