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A009236
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E.g.f. exp(sinh(x)^2) (even powers only).
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1
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1, 2, 20, 392, 12560, 579872, 36034880, 2874676352, 284538241280, 34058188677632, 4831480473359360, 799233222752602112, 152126941229960990720, 32947584100184816033792, 8042650107769696199720960, 2194728130327915760239542272, 664779526701915421094019399680
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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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a(n) = Sum_{k=1..n}(Sum_{i=0..2*k} (-1)^i*(k-i)^(2*n)*binomial(2*k,i))*(2^(2*(n-k)) /k!)), n>0, a(0)=1. - Vladimir Kruchinin, Jun 06 2011
a(n) = 4^n*Sum_{k=0..2*n} (-1)^k*binomial(2*n, k)*B(k, 1/4)*B(2*n-k, 1/4) where B(n, x) are the Bell polynomials. - Peter Luschny, Sep 10 2017
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MAPLE
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A009236 := n -> 4^n*(add(binomial(2*n, k)*(-1)^k*BellB(k, 1/4)*BellB(2*n-k, 1/4), k = 0..2*n)): seq(A009236(n), n=0..16); # Peter Luschny, Sep 10 2017
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(
b(n-2*j)*binomial(n-1, 2*j-1)*2^(2*j-1), j=1..n/2))
end:
a:= n-> b(2*n):
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MATHEMATICA
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b[n_] := b[n] = If[n == 0, 1, Sum[b[n-2*j]*Binomial[n-1, 2*j-1]*2^(2*j-1), {j, 1, n/2}]];
a[n_] := b[2*n];
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PROG
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(Maxima) a(n):=sum(sum((-1)^i*(k-i)^(2*n)*binomial(2*k, i), i, 0, 2*k)*(2^(2*(n-k))/k!), k, 1, n); /* Vladimir Kruchinin, Jun 06 2011 */
(PARI) my(x='x+O('x^40), v = Vec(serlaplace(exp(sinh(x)^2)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, May 21 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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