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A158757 Expansion of e.g.f. exp(t*x)/(1 - x^2/t^2 - t^3* x^3).

%I A158757 #13 Feb 06 2023 12:46:22
%S A158757 1,0,0,1,2,0,0,0,1,0,0,6,0,0,0,7,24,0,0,0,12,0,0,0,25,0,0,120,0,0,0,
%T A158757 260,0,0,0,61,720,0,0,0,360,0,0,0,1470,0,0,0,841,0,0,5040,0,0,0,15960,
%U A158757 0,0,0,5082,0,0,0,5251,40320,0,0,0,20160,0,0,0,122640,0,0,0,134456,0,0,0,20497
%N A158757 Expansion of e.g.f. exp(t*x)/(1 - x^2/t^2 - t^3* x^3).
%D A158757 H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973, page 221.
%H A158757 G. C. Greubel, <a href="/A158757/b158757.txt">Rows n = 0..50 of the irregular triangle, flattened</a>
%F A158757 T(n, k) = coefficients of e.g.f.: exp(t*x)/(1 - x^2/t^2 - t^3* x^3).
%F A158757 From _G. C. Greubel_, Dec 05 2021: (Start)
%F A158757 T(n, 2*n) = A330044(n).
%F A158757 T(n, 0) = A005359(n).
%F A158757 T(n, 2) = A005212(n). (End)
%e A158757 Irregular triangle begins as:
%e A158757       1;
%e A158757       0, 0,    1;
%e A158757       2, 0,    0, 0,   1;
%e A158757       0, 0,    6, 0,   0, 0,     7;
%e A158757      24, 0,    0, 0,  12, 0,     0, 0,   25;
%e A158757       0, 0,  120, 0,   0, 0,   260, 0,    0, 0,   61;
%e A158757     720, 0,    0, 0, 360, 0,     0, 0, 1470, 0,    0, 0, 841;
%e A158757       0, 0, 5040, 0,   0, 0, 15960, 0,    0, 0, 5082, 0,   0, 0, 5251;
%t A158757 Table[CoefficientList[Expand[t^n*n!*SeriesCoefficient[Series[Exp[t*x]/(1 - x^2/t^2 - t^3*x^3), {x, 0, 20}], n]], t], {n, 0, 10}]//Flatten
%o A158757 (Sage)
%o A158757 f(x,t) = exp(t*x)/(1 - x^2/t^2 - t^3*x^3)
%o A158757 def A158757(n,k): return ( factorial(n)*t^n*( f(x,t) ).series(x, 20).list()[n] ).series(t,2*n+1).list()[k]
%o A158757 flatten([[A158757(n,k) for k in (0..2*n)] for n in (0..10)]) # _G. C. Greubel_, Dec 05 2021
%Y A158757 Cf. A005212, A005359, A158706, A330044.
%K A158757 nonn,tabf
%O A158757 0,5
%A A158757 _Roger L. Bagula_, Mar 25 2009
%E A158757 Edited by _G. C. Greubel_, Dec 01 2021