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A240558
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a(n) = 2^n*n!/((floor(n/2)+1)*floor(n/2)!^2).
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2
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1, 2, 4, 24, 32, 320, 320, 4480, 3584, 64512, 43008, 946176, 540672, 14057472, 7028736, 210862080, 93716480, 3186360320, 1274544128, 48432676864, 17611882496, 739699064832, 246566354944, 11342052327424, 3489862254592, 174493112729600, 49855175065600
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OFFSET
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0,2
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LINKS
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FORMULA
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O.g.f.: ((i*(2*x*(8*x+1)-1))/sqrt(16*x^2-1)-2*x+1) /(8*x^2), where i=sqrt(-1).
For a recurrence see the Sage program.
a(n) = Sum_{k=0..n} A056040(n)*C(n,k)/(floor(n/2)+1).
a(n) = Sum_{k=0..n} n!*C(n,k)/((floor(n/2)+1)*(floor(n/2)!)^2).
a(n) = 2^n*n!*[x^n]((x+1)*hypergeom([],[2],x^2)).
a(n) ~ 2^(n+N)/((n+1)^<n>*sqrt(Pi*(2*N+1))); here <n> = 1 if n is even, 0 otherwise and N = n+<n>+1. (End)
Conjecture: -(n+2)*(n^2-5)*a(n) +8*(-2*n-1)*a(n-1) +16*(n-1)*(n^2+2*n-4)*a(n-2)=0. - R. J. Mathar, Jun 14 2016
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MAPLE
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A240558 := n -> 2^n*n!/((iquo(n, 2)+1)*iquo(n, 2)!^2):
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MATHEMATICA
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Table[SeriesCoefficient[((I*(2*x*(8*x+1)-1))/Sqrt[16*x^2-1]-2*x+1) /(8*x^2), {x, 0, n}], {n, 0, 22}]
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PROG
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(Sage)
x, n = 1, 1
while True:
yield x
m = 2*n if is_odd(n) else 8/(n+2)
x *= m
n += 1
a = A240558(); [next(a) for i in range(36)]
(PARI) x='x+O('x^50); Vec(round((I*(2*x*(8*x+1)-1))/sqrt(16*x^2-1)-2*x+1) /(8*x^2)) \\ G. C. Greubel, Apr 05 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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