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

%I #13 Feb 06 2023 12:46:22

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

%D H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973, page 221.

%H G. C. Greubel, <a href="/A158757/b158757.txt">Rows n = 0..50 of the irregular triangle, flattened</a>

%F T(n, k) = coefficients of e.g.f.: exp(t*x)/(1 - x^2/t^2 - t^3* x^3).

%F From _G. C. Greubel_, Dec 05 2021: (Start)

%F T(n, 2*n) = A330044(n).

%F T(n, 0) = A005359(n).

%F T(n, 2) = A005212(n). (End)

%e Irregular triangle begins as:

%e 1;

%e 0, 0, 1;

%e 2, 0, 0, 0, 1;

%e 0, 0, 6, 0, 0, 0, 7;

%e 24, 0, 0, 0, 12, 0, 0, 0, 25;

%e 0, 0, 120, 0, 0, 0, 260, 0, 0, 0, 61;

%e 720, 0, 0, 0, 360, 0, 0, 0, 1470, 0, 0, 0, 841;

%e 0, 0, 5040, 0, 0, 0, 15960, 0, 0, 0, 5082, 0, 0, 0, 5251;

%t 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 (Sage)

%o f(x,t) = exp(t*x)/(1 - x^2/t^2 - t^3*x^3)

%o 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 flatten([[A158757(n,k) for k in (0..2*n)] for n in (0..10)]) # _G. C. Greubel_, Dec 05 2021

%Y Cf. A005212, A005359, A158706, A330044.

%K nonn,tabf

%O 0,5

%A _Roger L. Bagula_, Mar 25 2009

%E Edited by _G. C. Greubel_, Dec 01 2021

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Last modified March 19 07:36 EDT 2024. Contains 370956 sequences. (Running on oeis4.)