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A024430 Expansion of e.g.f. cosh(exp(x)-1).

%I A024430 #60 Jul 03 2025 03:02:46
%S A024430 1,0,1,3,8,25,97,434,2095,10707,58194,338195,2097933,13796952,
%T A024430 95504749,692462671,5245040408,41436754261,340899165549,2915100624274,
%U A024430 25857170687507,237448494222575,2253720620740362,22078799199129799,222987346441156585,2319210969809731600
%N A024430 Expansion of e.g.f. cosh(exp(x)-1).
%C A024430 Number of partitions of an n-element set into an even number of classes.
%C A024430 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 A sequence (cf. A024429).
%D A024430 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 226, 5th line of table.
%D A024430 S. K. Ghosal, J. K. Mandal, Stirling Transform Based Color Image Authentication, Procedia Technology, 2013 Volume 10, 2013, Pages 95-104.
%D A024430 L. Lovasz, Combinatorial Problems and Exercises, North-Holland, 1993, pp. 15, 148.
%H A024430 Alois P. Heinz, <a href="/A024430/b024430.txt">Table of n, a(n) for n = 0..576</a>
%H A024430 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 A024430 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform.</a>
%F A024430 a(n) = S(n, 2) + S(n, 4) + ... + S(n, 2k), where k = [ n/2 ], S(i, j) are Stirling numbers of second kind.
%F A024430 E.g.f.: cosh(exp(x)-1). - _N. J. A. Sloane_, Jan 28 2001
%F A024430 a(n) = (A000110(n) + A000587(n)) / 2. - _Peter Luschny_, Apr 25 2011
%F A024430 O.g.f.: Sum_{n>=0} x^(2*n) / Product_{k=0..2*n} (1 - k*x). - _Paul D. Hanna_, Sep 05 2012
%F A024430 G.f.: G(0)/(1+x) where G(k) = 1 - x*(2*k+1)/((2*x*k-1) - x*(2*x*k-1)/(x - (2*k+1)*(2*x*k+x-1)/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 05 2013
%F A024430 G.f.: G(0)/(1+2*x) where G(k) = 1 - 2*x*(k+1)/((2*x*k-1) - x*(2*x*k-1)/(x - 2*(k+1)*(2*x*k+x-1)/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 05 2013
%F A024430 a(n) ~ n^n / (2 * (LambertW(n))^n * exp(n+1-n/LambertW(n)) * sqrt(1+LambertW(n))). - _Vaclav Kotesovec_, Aug 04 2014
%p A024430 b:= proc(n, t) option remember; `if`(n=0, t, add(
%p A024430        b(n-j, 1-t)*binomial(n-1, j-1), j=1..n))
%p A024430     end:
%p A024430 a:= n-> b(n, 1):
%p A024430 seq(a(n), n=0..28);  # _Alois P. Heinz_, Jan 15 2018
%p A024430 with(combinat); seq((bell(n) + BellB(n, -1))/2, n = 0..20); # _G. C. Greubel_, Oct 09 2019
%t A024430 nn=20;a=Exp[Exp[x]-1];Range[0,nn]!CoefficientList[Series[(a+1/a)/2,{x,0,nn}],x]  (* _Geoffrey Critzer_, Nov 04 2012 *)
%t A024430 Table[(BellB[n] + BellB[n, -1])/2, {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 01 2015 *)
%o A024430 (Sage)
%o A024430 def A024430(n) :
%o A024430     return add(stirling_number2(n,i) for i in range(0,n+(n+1)%2,2))
%o A024430 # _Peter Luschny_, Feb 28 2012
%o A024430 (PARI) {a(n)=polcoeff(sum(m=0, n, x^(2*m)/prod(k=1, 2*m, 1-k*x +x*O(x^n))), n)} \\ _Paul D. Hanna_, Sep 05 2012
%o A024430 (Magma) a:= func< n | (&+[StirlingSecond(n,2*k): k in [0..Floor(n/2)]]) >;
%o A024430 [a(n): n in [0..25]]; // _G. C. Greubel_, Oct 09 2019
%o A024430 (GAP) List([0..25], n-> Sum([0..Int(n/2)], k-> Stirling2(n,2*k)) ); # _G. C. Greubel_, Oct 09 2019
%Y A024430 Cf. A024429, A121867, A121868, A000110, A000587.
%K A024430 nonn
%O A024430 0,4
%A A024430 _Clark Kimberling_
%E A024430 Description changed by _N. J. A. Sloane_, Jun 14 2003 and again Sep 05 2006