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A205571 Expansion of e.g.f. 1/(1 - x*cosh(x)).

%I A205571 #34 Apr 19 2025 06:57:39
%S A205571 1,1,2,9,48,305,2400,22057,230272,2708001,35412480,509177801,
%T A205571 7986468864,135718942801,2483729876992,48699677975145,
%U A205571 1018542257111040,22634000289407297,532557637644976128,13226748101381102473,345792863300174479360,9492229607399841038961
%N A205571 Expansion of e.g.f. 1/(1 - x*cosh(x)).
%C A205571 Radius of convergence of e.g.f. is |x| < r where r = 0.7650099545507... satisfies cosh(r) = 1/r. See A069814.
%H A205571 Vincenzo Librandi, <a href="/A205571/b205571.txt">Table of n, a(n) for n = 0..200</a>
%F A205571 a(2*n-1) == 1 (mod 4), a(2*n+2) == 0 (mod 4), for n>=1.
%F A205571 a(n) ~ n!/(1+r*sqrt(1-r^2))*(1/r)^n, where r = A069814 = 0.7650099545507321... is the root of the equation r*cosh(r)=1. - _Vaclav Kotesovec_, Feb 13 2013
%F A205571 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1) * (2*k+1) * a(n-2*k-1). - _Ilya Gutkovskiy_, Mar 10 2022
%F A205571 a(n) = Sum_{k=0..n} k! * A185951(n,k). - _Seiichi Manyama_, Feb 17 2025
%e A205571 E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 48*x^4/4! + 305*x^5/5! +...
%t A205571 CoefficientList[Series[1/(1-x*Cosh[x]), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Feb 13 2013 *)
%o A205571 (PARI) {a(n)=n!*polcoeff(1/(1-x*cosh(x +x*O(x^n))),n)}
%o A205571 (PARI) a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
%o A205571 a(n) = sum(k=0, n, k!*a185951(n, k)); \\ _Seiichi Manyama_, Feb 17 2025
%Y A205571 Cf. A069814, A185951.
%Y A205571 Cf. A000111, A006154, A352252.
%Y A205571 Cf. A381206, A381207.
%K A205571 nonn
%O A205571 0,3
%A A205571 _Paul D. Hanna_, Jan 28 2012