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A331326
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a(n) = n!*[x^n] sinh(x/(1 - x))/(1 - x).
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2
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0, 1, 4, 19, 112, 801, 6756, 65563, 717760, 8729857, 116570980, 1693096131, 26548383984, 446689827169, 8023582921732, 153192673528651, 3097301219335936, 66095983547942913, 1484384376886189380, 34991710162280602867, 863797053818651591920, 22282392569877969167521
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listen;
history;
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n/2} |A021009(n, 2*k+1)|.
a(n) = Sum_{k=0..n} binomial(n, 2*k+1)*n!/(2*k+1)!.
a(n) = n*n!*hypergeom([1/2 - n/2, 1 - n/2], [1, 3/2, 3/2], 1/4).
(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
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
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
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MAPLE
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gf := sinh(x/(1 - x))/(1 - x): ser := series(gf, x, 22):
seq(n!*coeff(ser, x, n), n=0..20);
# Alternative: seq(add(abs(A021009(n, 2*k+1)), k=0..n/2), n=0..21);
A331326 := proc(n) local S; S := proc(n, k) option remember; `if`(k = 0, 1,
`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:
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MATHEMATICA
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a[n_] := n n! HypergeometricPFQ[{1/2 - n/2, 1 - n/2}, {1, 3/2, 3/2}, 1/4];
Array[a, 22, 0]
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PROG
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(PARI) x='x+O('x^22); concat(0, Vec(serlaplace(sinh(x/(1-x))/(1-x))))
(Python)
sa, sb, ta, tb, n = 1, 2, 1, 0, 2
yield 0
yield ta
while(True):
s = 2*n*sb - ((n-1)**2)*sa
t = 2*(n-1)*tb - ((n-1)**2)*ta
sa, sb, ta, tb = sb, s, tb, t
n += 1
yield (s - t)//2
a = A331326(); print([next(a) for _ in range(22)])
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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