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A024429
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Expansion of e.g.f. sinh(exp(x)-1).
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39
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0, 1, 1, 2, 7, 27, 106, 443, 2045, 10440, 57781, 340375, 2115664, 13847485, 95394573, 690495874, 5235101739, 41428115543, 341177640610, 2917641580783, 25866987547865, 237421321934176, 2252995117706961, 22073206655954547, 222971522853648704, 2319379362420267753
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OFFSET
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0,4
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COMMENTS
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Number of partitions of an n-element set into an odd number of classes. - Peter Luschny, Apr 25 2011
Let A(0) = 1, B(0) = 0; A(n+1) = Sum_{k=0..n} binomial(n,k)*B(k), B(n+1) = Sum_{k=0..n} binomial(n,k)*A(k); entry gives B sequence (cf. A024430).
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 226, 4th line of table.
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LINKS
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FORMULA
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S(n,1) + S(n,3) + ... + S(n,2k+1), where k = [ (n-1)/2 ] and S(i,j) are Stirling numbers of second kind.
G.f.: x*G(0) where G(k) = 1 - x*(2*k+1)/((2*x*k+x-1) - x*(2*x*k+x-1)/(x - (2*k+1)*(2*x*k+2*x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 06 2013.
G.f.: x*G(0)/(1+x) where G(k) = 1 - 2*x*(k+1)/((2*x*k+x-1) - x*(2*x*k+x-1)/(x - 2*(k+1)*(2*x*k+2*x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 06 2013.
G.f.: -x*(1+x)*sum(k=>0 x^(2*k)/((2*x*k+x-1)*prod(p=0...k (2*x*p-1)*(2*x*p-x-1)) . - Sergei N. Gladkovskii, Jan 06 2013
a(n) ~ n^n / (2 * (LambertW(n))^n * exp(n+1-n/LambertW(n)) * sqrt(1+LambertW(n))). - Vaclav Kotesovec, Aug 04 2014
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EXAMPLE
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G.f. = x + x^2 + 2*x^3 + 7*x^4 + 27*x^5 + 106*x^6 + 443*x^7 + 2045*x^8 + ...
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MAPLE
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b:= proc(n, t) option remember; `if`(n=0, t, add(
b(n-j, 1-t)*binomial(n-1, j-1), j=1..n))
end:
a:= n-> b(n, 0):
with(combinat); seq((bell(n) - BellB(n, -1))/2, n = 0..25); # G. C. Greubel, Oct 09 2019
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MATHEMATICA
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CoefficientList[Series[Sinh[E^x-1], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 04 2014 *)
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PROG
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(Sage)
return add(stirling_number2(n, i) for i in range(1, n+n%2, 2))
(PARI) x='x+O('x^50); concat([0], Vec(serlaplace(sinh(exp(x)-1)))) \\ G. C. Greubel, Nov 12 2017
(Magma) a:= func< n | (&+[StirlingSecond(n, 2*k+1): k in [0..Floor(n/2)]]) >;
(GAP) List([0..25], n-> Sum([0..Int(n/2)], k-> Stirling2(n, 2*k+1)) ); # G. C. Greubel, Oct 09 2019
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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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