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%I #14 Feb 05 2023 09:23:14
%S 1,0,0,0,-2,0,6,0,-24,0,120,40,0,-720,-420,0,5040,3948,0,-40320,
%T -38304,-2240,0,362880,396576,50400
%N Triangular sequence from expansion coefficients of asymptotic Hermite Polynomial from Roman: p(x,t)= (1 + t)^(-x)*Exp[x*(t - t^2/2)].
%C Absolute values of row sums give A038205.
%D Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 130.
%F p(x,t)= (1 + t)^(-x)*Exp[x*(t - t^2/2)]=Sum[q(x,n)*t^n/n!,{n,0,Infinity}]; out(n,m)=n!*Coefficients(q(x,n)).
%e {1},
%e {0},
%e {0},
%e {0, -2},
%e {0, 6},
%e {0, -24},
%e {0, 120, 40},
%e {0, -720, -420},
%e {0, 5040, 3948},
%e {0, -40320, -38304, -2240},
%e {0, 362880, 396576, 50400}
%t p[t_] := (1 + t)^(-x)*Exp[x*(t - t^2/2)];
%t Table[ ExpandAll[n!SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}];
%t a = Table[n!* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}];
%t Flatten[{{1}, {0}, {0}, {0, -2}, {0, 6}, {0, -24}, {0, 120, 40}, {0, -720, -420}, {0, 5040, 3948}, {0, -40320, -38304, -2240}, {0, 362880, 396576, 50400}}]
%Y Cf. A038205, A137286.
%K uned,tabf,sign
%O 1,5
%A _Roger L. Bagula_, Apr 21 2008
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