A331325 - OEIS

%I #22 Feb 17 2024 03:40:03

%S 1,1,3,15,97,745,6571,65359,723969,8842257,118091251,1712261551,

%T 26786070433,449634481465,8059974923547,153634497337455,

%U 3102367733191681,66145005096272929,1484586887025099619,34983117545622446287,863397428225495045601,22269844592814969946761

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

%H Robert Israel, <a href="/A331325/b331325.txt">Table of n, a(n) for n = 0..443</a>

%F a(n) + A331326(n) = A002720(n).

%F a(n) - A331326(n) = A009940(n).

%F a(n) = Sum_{k=0..n/2} |A021009(n, 2*k)|.

%F a(n) = Sum_{k=0..n} binomial(n, 2*k)*n!/(2*k)!.

%F a(n) = n!*hypergeom([1/2 - n/2, -n/2], [1/2, 1/2, 1], 1/4).

%F (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 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 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 gf := cosh(x/(1 - x))/(1 - x): ser := series(gf, x, 22):

%p seq(n!*coeff(ser, x, n), n=0..21);

%p # Alternative: seq(add(abs(A021009(n, 2*k)), k=0..n/2), n=0..21);

%p A331325 := proc(n) local S; S := proc(n, k) option remember; `if`(k = 0, 1,

%p `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 seq(A331325(n), n=0..21);

%t a[n_] := n! HypergeometricPFQ[{1/2 - n/2, -n/2}, {1, 1/2, 1/2}, 1/4];

%t Array[a, 22, 0]

%o (PARI) x='x+O('x^22); Vec(serlaplace(cosh(x/(1-x))/(1-x)))

%o (Python)

%o def A331325():

%o     sa, sb, ta, tb, n = 1, 2, 1, 0, 2

%o     yield sa

%o     yield ta

%o     while(True):

%o         s = 2*n*sb - ((n-1)**2)*sa

%o         t = 2*(n-1)*tb - ((n-1)**2)*ta

%o         sa, sb, ta, tb = sb, s, tb, t

%o         n += 1

%o         yield (s + t)//2

%o a = A331325(); print([next(a) for _ in range(22)])

%Y Cf. A002720, A009940, A021009, A331326.

%K nonn

%O 0,3

%A _Peter Luschny_, Jan 21 2020