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A009229
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Expansion of e.g.f. exp(sinh(x)*cosh(x)).
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4
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1, 1, 1, 5, 17, 57, 337, 1661, 9697, 68529, 451809, 3570613, 29324913, 247557481, 2348275633, 22399292077, 229227939521, 2476740370529, 27378158737089, 323000223133285, 3912000089144273, 49350756905258905, 649905519911431953, 8776449485870211549
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OFFSET
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0,4
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LINKS
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FORMULA
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a(n) = sum(k=1..n, ((sum(i=0..k, (-1)^i*(k-2*i)^n*binomial(k,i)))*2^(n-2*k))/k!), n>0, a(0)=1. - Vladimir Kruchinin, May 29 2011
a(n) ~ exp(cosh(r)*sinh(r)-n) * n^(n+1/2) / (sqrt(cosh(2*r) + 2*r*sinh(2*r)) * r^(n+1/2)), where r is the root of the equation r*(cosh(r)^2 + sinh(r)^2) = n.
(a(n)/n!)^(1/n) ~ 2*(exp(1/LambertW(4*n))/LambertW(4*n)).
(End)
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * 4^k * a(n-2*k-1). - Ilya Gutkovskiy, Feb 24 2022
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MAPLE
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a:= proc(n) option remember; `if`(n=0, 1, add(`if`(j::odd,
a(n-j)*binomial(n-1, j-1)*2^(j-1), 0), j=1..n))
end:
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MATHEMATICA
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With[{nn=30}, CoefficientList[Series[Exp[Sinh[x]Cosh[x]], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Sep 23 2011 *)
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PROG
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(Maxima)
a(n):=sum(((sum((-1)^i*(k-2*i)^n*binomial(k, i), i, 0, k))*2^(n-2*k))/k!, k, 1, n); /* Vladimir Kruchinin, May 29 2011 */
(PARI) x='x+O('x^66); /* that many terms */
Vec(serlaplace(exp(sinh(x)*cosh(x)))) /* show terms */ /* Joerg Arndt, May 29 2011 */
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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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EXTENSIONS
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STATUS
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approved
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