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%I #21 Feb 15 2023 10:59:13
%S 1,1,1,4,25,96,481,3368,20721,141760,1146721,9098112,77652169,
%T 726208640,6891125697,69344336896,738718169569,8076031881216,
%U 92647353941569,1106883171037184,13616813607795321,174298975125127168,2304515271134124577
%N Expansion of e.g.f. (exp(x*exp(x)) + exp(x/exp(x)))/2.
%H Seiichi Manyama, <a href="/A195509/b195509.txt">Table of n, a(n) for n = 0..541</a>
%F E.g.f.: Sum_{n>=0} x^n*cosh(n*x)/n!.
%F a(n) = Sum_{m=0..floor((n-1)/2)} binomial(n,2*m)*(n-2*m)^(2*m) for n >= 1. - _Vladimir Kruchinin_, Mar 10 2013 [Edited by _Petros Hadjicostas_, May 06 2020]
%e E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 25*x^4/4! + 96*x^5/5! + 481*x^6/6! +...
%p a := proc(n) local m: add(binomial(n, 2*m)*(n - 2*m)^(2*m), m = 0 .. floor(1/2*n - 1/2)): end proc:
%p seq(a(n), n = 1..30); # _Petros Hadjicostas_, May 06 2020 (for n >= 1)
%o (PARI) {a(n)=local(X=x+x*O(x^n),A=1+X);A=(exp(X*exp(X))+exp(X/exp(X)))/2;n!*polcoeff(A,n)}
%o (PARI) {a(n)=n!*polcoeff(sum(m=0,n,x^m*cosh(m*x+x*O(x^n))/m!),n)}
%o (PARI) a(n) = sum(k=0, n\2, (n-2*k)^(2*k)*binomial(n, 2*k)); \\ _Seiichi Manyama_, Feb 15 2023
%Y Cf. A218296, A277464.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Sep 19 2011
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