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%I #23 Feb 17 2024 03:39:41
%S 0,1,4,19,112,801,6756,65563,717760,8729857,116570980,1693096131,
%T 26548383984,446689827169,8023582921732,153192673528651,
%U 3097301219335936,66095983547942913,1484384376886189380,34991710162280602867,863797053818651591920,22282392569877969167521
%N a(n) = n!*[x^n] sinh(x/(1 - x))/(1 - x).
%H Robert Israel, <a href="/A331326/b331326.txt">Table of n, a(n) for n = 0..443</a>
%F a(n) + A331325(n) = A002720(n).
%F A331325(n) - a(n) = A009940(n).
%F a(n) = Sum_{k=0..n/2} |A021009(n, 2*k+1)|.
%F a(n) = Sum_{k=0..n} binomial(n, 2*k+1)*n!/(2*k+1)!.
%F a(n) = n*n!*hypergeom([1/2 - n/2, 1 - n/2], [1, 3/2, 3/2], 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 22 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 17 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 := sinh(x/(1 - x))/(1 - x): ser := series(gf, x, 22):
%p seq(n!*coeff(ser, x, n), n=0..20);
%p # Alternative: seq(add(abs(A021009(n, 2*k+1)), k=0..n/2), n=0..21);
%p A331326 := 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+1), k=0..n) end:
%p seq(A331326(n), n=0..21);
%t a[n_] := n n! HypergeometricPFQ[{1/2 - n/2, 1 - n/2}, {1, 3/2, 3/2}, 1/4];
%t Array[a, 22, 0]
%o (PARI) x='x+O('x^22); concat(0,Vec(serlaplace(sinh(x/(1-x))/(1-x))))
%o (Python)
%o def A331326():
%o sa, sb, ta, tb, n = 1, 2, 1, 0, 2
%o yield 0
%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 = A331326(); print([next(a) for _ in range(22)])
%Y Cf. A002720, A009940, A021009, A331325.
%K nonn
%O 0,3
%A _Peter Luschny_, Jan 21 2020
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