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 A024430 Expansion of e.g.f. cosh(exp(x)-1). 15

%I

%S 1,0,1,3,8,25,97,434,2095,10707,58194,338195,2097933,13796952,

%T 95504749,692462671,5245040408,41436754261,340899165549,2915100624274,

%U 25857170687507,237448494222575,2253720620740362,22078799199129799,222987346441156585,2319210969809731600

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

%C Number of partitions of an n-element set into an even number of classes.

%C 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 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 226, 5th line of table.

%D S. K. Ghosal, J. K. Mandal, Stirling Transform Based Color Image Authentication, Procedia Technology, 2013 Volume 10, 2013, Pages 95-104.

%D L. Lovasz, Combinatorial Problems and Exercises, North-Holland, 1993, pp. 15, 148.

%H Alois P. Heinz, <a href="/A024430/b024430.txt">Table of n, a(n) for n = 0..250</a>

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform.</a>

%F 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 E.g.f.: cosh(exp(x)-1). - _N. J. A. Sloane_, Jan 28, 2001

%F a(n) = (A000110(n) + A000587(n)) / 2. - _Peter Luschny_, Apr 25 2011

%F O.g.f.: Sum_{n>=0} x^(2*n) / Product_{k=0..2*n} (1 - k*x). - _Paul D. Hanna_, Sep 05 2012

%F 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 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 a(n) ~ n^n / (2 * (LambertW(n))^n * exp(n+1-n/LambertW(n)) * sqrt(1+LambertW(n))). - _Vaclav Kotesovec_, Aug 04 2014

%p b:= proc(n, t) option remember; `if`(n=0, t, add(

%p b(n-j, 1-t)*binomial(n-1, j-1), j=1..n))

%p end:

%p a:= n-> b(n, 1):

%p seq(a(n), n=0..28); # _Alois P. Heinz_, Jan 15 2018

%t 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 Table[(BellB[n] + BellB[n, -1])/2, {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 01 2015 *)

%o (Sage)

%o def A024430(n) :

%o return add(stirling_number2(n,i) for i in range(0,n+(n+1)%2,2))

%o # _Peter Luschny_, Feb 28 2012

%o (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

%Y Cf. A024429, A121867, A121868, A000110, A000587.

%K nonn

%O 0,4

%A _Clark Kimberling_

%E Description changed by _N. J. A. Sloane_, Jun 14 2003 and again Sep 05 2006

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Last modified June 17 07:00 EDT 2019. Contains 324183 sequences. (Running on oeis4.)