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A137514 A triangular sequence from umbral calculus expansion of Simon Plouffe's rational polynomial for A002890: p(x,t) = exp(x*t)*(1 - 6*t + 9*t^2 - 4*t^3 + t^4)/(4*t - 1)/(2*t - 1).

%I A137514 #10 May 06 2025 21:52:23
%S A137514 1,0,1,2,0,1,12,6,0,1,120,48,12,0,1,1680,600,120,20,0,1,31680,10080,
%T A137514 1800,240,30,0,1,766080,221760,35280,4200,420,42,0,1,22579200,6128640,
%U A137514 887040,94080,8400,672,56,0,1,778014720,203212800,27578880,2661120,211680,15120,1008,72,0,1
%N A137514 A triangular sequence from umbral calculus expansion of _Simon Plouffe_'s rational polynomial for A002890: p(x,t) = exp(x*t)*(1 - 6*t + 9*t^2 - 4*t^3 + t^4)/(4*t - 1)/(2*t - 1).
%C A137514 Row sums:
%C A137514 {1, 1, 3, 19, 181, 2421, 43831, 1027783, 29698089, 1011695401, 39319102891}
%C A137514 The t's here are actually Sqrt[] of the variables that give Gamma(1,t) in the Hill reference and is the expansion of Plouffe's rational polynomial for A002890. So this result is related closely to Hill's Gamma(x,y) and seems to be a generalization of the A002890 polynomial.
%D A137514 Terrel L. Hill, Statistical Mechanics: Principles and Selected Applications, Dover, New York, 1956, page 336 ff
%F A137514 p(x,t) = exp(x*t)*(1 - 6*t + 9*t^2 - 4*t^3 + t^4)/(4*t - 1)/(2*t - 1) = Sum_{n>=0} P(x,n)*t^n/n!; out_n,m=n!*Coefficients(P(x,n)).
%e A137514 Triangle begins:
%e A137514   {1},
%e A137514   {0, 1},
%e A137514   {2, 0, 1},
%e A137514   {12, 6, 0, 1},
%e A137514   {120, 48, 12, 0, 1},
%e A137514   {1680, 600, 120, 20, 0, 1},
%e A137514   {31680, 10080, 1800, 240, 30, 0, 1},
%e A137514   {766080, 221760, 35280, 4200, 420, 42, 0, 1},
%e A137514   {22579200, 6128640, 887040, 94080, 8400, 672, 56, 0, 1},
%e A137514   {778014720, 203212800, 27578880, 2661120, 211680, 15120, 1008, 72, 0, 1},
%e A137514   ...
%t A137514 Clear[p, f, g] p[t_] = Exp[x*t]*(1 - 6*t + 9*t^2 - 4*t^3 + t^4)/(4*t - 1)/(2*t - 1); Table[ ExpandAll[n!*SeriesCoefficient[Series[p[t], {t, 0, 30}], n]], {n, 0, 10}] a = Table[ CoefficientList[n!*SeriesCoefficient[; FullSimplify[Series[p[t], {t, 0, 30}]], n], x], {n, 0, 10}]; Flatten[a]
%Y A137514 Cf. A002890, A136264.
%K A137514 nonn,uned,tabl
%O A137514 1,4
%A A137514 _Roger L. Bagula_, Apr 23 2008