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A024429 Expansion of e.g.f. sinh(exp(x)-1).

%I A024429 #63 Jul 03 2025 03:02:30
%S A024429 0,1,1,2,7,27,106,443,2045,10440,57781,340375,2115664,13847485,
%T A024429 95394573,690495874,5235101739,41428115543,341177640610,2917641580783,
%U A024429 25866987547865,237421321934176,2252995117706961,22073206655954547,222971522853648704,2319379362420267753
%N A024429 Expansion of e.g.f. sinh(exp(x)-1).
%C A024429 Number of partitions of an n-element set into an odd number of classes. - _Peter Luschny_, Apr 25 2011
%C A024429 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).
%D A024429 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 226, 4th line of table.
%H A024429 G. C. Greubel, <a href="/A024429/b024429.txt">Table of n, a(n) for n = 0..400</a>
%H A024429 A. Fekete and G. Martin, <a href="http://www.jstor.org/stable/2695545">Problem 10791: Squared Series Yielding Integers</a>, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178.
%H A024429 S. K. Ghosal, J. K. Mandal, <a href="https://doi.org/10.1016/j.protcy.2013.12.341">Stirling Transform Based Color Image Authentication</a>, Procedia Technology, 2013 Volume 10, 2013, Pages 95-104.
%H A024429 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform.</a>
%F A024429 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.
%F A024429 E.g.f.: sinh(exp(x)-1). - _N. J. A. Sloane_, Jan 28 2001
%F A024429 a(n) = (A000110(n) - A000587(n)) / 2. - _Peter Luschny_, Apr 25 2011
%F A024429 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
%F A024429 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
%F A024429 G.f.: -x*(1+x)*Sum_{k>=0} x^(2*k)/((2*x*k+x-1)*Product_{p=0..k} (2*x*p-1)*(2*x*p-x-1)). - _Sergei N. Gladkovskii_, Jan 06 2013
%F A024429 G.f.: Sum_{k>=0} x^(2*k+1)/(Product_{i=0..2*k+1} 1-i*x). - _Sergei N. Gladkovskii_, Jan 06 2013
%F A024429 a(n) ~ n^n / (2 * (LambertW(n))^n * exp(n+1-n/LambertW(n)) * sqrt(1+LambertW(n))). - _Vaclav Kotesovec_, Aug 04 2014
%e A024429 G.f. = x + x^2 + 2*x^3 + 7*x^4 + 27*x^5 + 106*x^6 + 443*x^7 + 2045*x^8 + ...
%p A024429 b:= proc(n, t) option remember; `if`(n=0, t, add(
%p A024429        b(n-j, 1-t)*binomial(n-1, j-1), j=1..n))
%p A024429     end:
%p A024429 a:= n-> b(n, 0):
%p A024429 seq(a(n), n=0..28);  # _Alois P. Heinz_, Jan 15 2018
%p A024429 with(combinat); seq((bell(n) - BellB(n, -1))/2, n = 0..25); # _G. C. Greubel_, Oct 09 2019
%t A024429 CoefficientList[Series[Sinh[E^x-1], {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Aug 04 2014 *)
%t A024429 Table[(BellB[n] - BellB[n, -1])/2, {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 01 2015 *)
%o A024429 (Sage)
%o A024429 def A024429(n) :
%o A024429     return add(stirling_number2(n,i) for i in range(1,n+n%2,2))
%o A024429 # _Peter Luschny_, Feb 28 2012
%o A024429 (PARI) x='x+O('x^50); concat([0], Vec(serlaplace(sinh(exp(x)-1)))) \\ _G. C. Greubel_, Nov 12 2017
%o A024429 (Magma) a:= func< n | (&+[StirlingSecond(n,2*k+1): k in [0..Floor(n/2)]]) >;
%o A024429 [a(n): n in [0..25]]; // _G. C. Greubel_, Oct 09 2019
%o A024429 (GAP) List([0..25], n-> Sum([0..Int(n/2)], k-> Stirling2(n,2*k+1)) ); # _G. C. Greubel_, Oct 09 2019
%Y A024429 Cf. A024430, A121867, A121868, A000110, A000587.
%K A024429 nonn
%O A024429 0,4
%A A024429 _Clark Kimberling_
%E A024429 Description changed by _N. J. A. Sloane_, Sep 05 2006