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A331325 a(n) = n!*[x^n] cosh(x/(1-x))/(1-x).

%I A331325 #22 Feb 17 2024 03:40:03
%S A331325 1,1,3,15,97,745,6571,65359,723969,8842257,118091251,1712261551,
%T A331325 26786070433,449634481465,8059974923547,153634497337455,
%U A331325 3102367733191681,66145005096272929,1484586887025099619,34983117545622446287,863397428225495045601,22269844592814969946761
%N A331325 a(n) = n!*[x^n] cosh(x/(1-x))/(1-x).
%H A331325 Robert Israel, <a href="/A331325/b331325.txt">Table of n, a(n) for n = 0..443</a>
%F A331325 a(n) + A331326(n) = A002720(n).
%F A331325 a(n) - A331326(n) = A009940(n).
%F A331325 a(n) = Sum_{k=0..n/2} |A021009(n, 2*k)|.
%F A331325 a(n) = Sum_{k=0..n} binomial(n, 2*k)*n!/(2*k)!.
%F A331325 a(n) = n!*hypergeom([1/2 - n/2, -n/2], [1/2, 1/2, 1], 1/4).
%F A331325 (n+1)^2*(n+2)^2*a(n) - 4*(n+2)^3*a(n+1) + (6*n^2+30*n+37)*a(n+2) - 4*(n+3)*a(n+3)+a(n+4)=0. - _Robert Israel_, Jan 23 2020
%F A331325 Sum_{n>=0} a(n) * x^n / (n!)^2 = (1/2) * exp(x) * (BesselI(0,2*sqrt(x)) + BesselJ(0,2*sqrt(x))). - _Ilya Gutkovskiy_, Jul 18 2020
%F A331325 a(n) ~ 2^(-3/2) * exp(2*sqrt(n)-n-1/2) * n^(n+1/4) * (1 + 31/(48*sqrt(n))). - _Vaclav Kotesovec_, Feb 17 2024
%p A331325 gf := cosh(x/(1 - x))/(1 - x): ser := series(gf, x, 22):
%p A331325 seq(n!*coeff(ser, x, n), n=0..21);
%p A331325 # Alternative: seq(add(abs(A021009(n, 2*k)), k=0..n/2), n=0..21);
%p A331325 A331325 := proc(n) local S; S := proc(n, k) option remember; `if`(k = 0, 1,
%p A331325 `if`(k > n, 0, S(n-1, k-1)/k + S(n-1, k))) end: n!*add(S(n, 2*k), k=0..n) end:
%p A331325 seq(A331325(n), n=0..21);
%t A331325 a[n_] := n! HypergeometricPFQ[{1/2 - n/2, -n/2}, {1, 1/2, 1/2}, 1/4];
%t A331325 Array[a, 22, 0]
%o A331325 (PARI) x='x+O('x^22); Vec(serlaplace(cosh(x/(1-x))/(1-x)))
%o A331325 (Python)
%o A331325 def A331325():
%o A331325     sa, sb, ta, tb, n = 1, 2, 1, 0, 2
%o A331325     yield sa
%o A331325     yield ta
%o A331325     while(True):
%o A331325         s = 2*n*sb - ((n-1)**2)*sa
%o A331325         t = 2*(n-1)*tb - ((n-1)**2)*ta
%o A331325         sa, sb, ta, tb = sb, s, tb, t
%o A331325         n += 1
%o A331325         yield (s + t)//2
%o A331325 a = A331325(); print([next(a) for _ in range(22)])
%Y A331325 Cf. A002720, A009940, A021009, A331326.
%K A331325 nonn
%O A331325 0,3
%A A331325 _Peter Luschny_, Jan 21 2020