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A322544 a(n) is the reciprocal of the coefficient of x^n in the power series defined by ((1+2x)*exp(x) + 3*exp(-x) - 4)/ (4x^2).

%I A322544 #45 Aug 15 2025 05:58:43
%S A322544 1,6,8,60,180,1680,8064,90720,604800,7983360,68428800,1037836800,
%T A322544 10897286400,186810624000,2324754432000,44460928512000,
%U A322544 640237370572800,13516122267648000,221172909834240000,5109094217170944000,93666727314800640000,2350183339898634240000,47726800133326110720000
%N A322544 a(n) is the reciprocal of the coefficient of x^n in the power series defined by ((1+2x)*exp(x) + 3*exp(-x) - 4)/ (4x^2).
%H A322544 Muniru A Asiru, <a href="/A322544/b322544.txt">Table of n, a(n) for n = 0..445</a>
%F A322544 a(n) = (n+2)!/(3*floor(n/2)-n+2).
%F A322544 a(n) = (4*(n+2)!)/(2n+5+3*(-1)^n).
%F A322544 a(n) = 4/([x^n]((exp(x)*(1+2x)+3*exp(-x)-4)/x^2)).
%F A322544 a(n) = (n+2)!/(A028242(n)+1).
%F A322544 a(n) = (n+2)!/A030451(n+1).
%F A322544 a(n) ~ sqrt(Pi/2)/72*exp(-n)*n^(n-1/2)*(1705 - 264*n + 288*n^2). - _Stefano Spezia_, Aug 11 2025
%F A322544 Sum_{n>=0} 1/a(n) = 3*cosh(1)/2 - 1. - _Amiram Eldar_, Aug 15 2025
%p A322544 a:=n->factorial(n+2)/(3*floor(n/2)-n+2): seq(a(n),n=0..25); # _Muniru A Asiru_, Dec 20 2018
%t A322544 Table[4*Factorial[n + 2]/(2*n + 5 + 3*(-1)^n), {n, 0, 25}]
%t A322544 (* or *)
%t A322544 Function[x, 1/x] /@
%t A322544 CoefficientList[Series[(Exp[x]/4 + 3/4*Exp[-x] + x/2*Exp[x] - 1)/x^2, {x, 0, 20}], x]
%o A322544 (PARI) a(n)={(4*(n+2)!)/(5 + 3*(-1)^n + 2*n)} \\ _Andrew Howroyd_, Dec 14 2018
%o A322544 (PARI) my(x='x + O('x^30)); Vec(apply(x->1/x, ((1+2*x)*exp(x) + 3*exp(-x) - 4)/ (4*x^2))) \\ _Michel Marcus_, Dec 19 2018
%o A322544 (GAP) List([0..25],n->(4*Factorial(n+2))/(2*n+5+3*(-1)^n)); # _Muniru A Asiru_, Dec 20 2018
%Y A322544 Cf. A060593 (even bisection, shifted), A028242 (denominator minus 1), A030451 (denominator, shifted), A107991 (Expansion of a similar function), A073743.
%K A322544 nonn
%O A322544 0,2
%A A322544 _Pierre-Alain Sallard_, Dec 14 2018