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 A024430 Expansion of e.g.f. cosh(exp(x)-1). 32
 1, 0, 1, 3, 8, 25, 97, 434, 2095, 10707, 58194, 338195, 2097933, 13796952, 95504749, 692462671, 5245040408, 41436754261, 340899165549, 2915100624274, 25857170687507, 237448494222575, 2253720620740362, 22078799199129799, 222987346441156585, 2319210969809731600 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Number of partitions of an n-element set into an even number of classes. 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). REFERENCES L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 226, 5th line of table. S. K. Ghosal, J. K. Mandal, Stirling Transform Based Color Image Authentication, Procedia Technology, 2013 Volume 10, 2013, Pages 95-104. L. Lovasz, Combinatorial Problems and Exercises, North-Holland, 1993, pp. 15, 148. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..576 A. Fekete and G. Martin, Problem 10791: Squared Series Yielding Integers, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178. Eric Weisstein's World of Mathematics, Stirling Transform. FORMULA a(n) = S(n, 2) + S(n, 4) + ... + S(n, 2k), where k = [ n/2 ], S(i, j) are Stirling numbers of second kind. E.g.f.: cosh(exp(x)-1). - N. J. A. Sloane, Jan 28, 2001 a(n) = (A000110(n) + A000587(n)) / 2. - Peter Luschny, Apr 25 2011 O.g.f.: Sum_{n>=0} x^(2*n) / Product_{k=0..2*n} (1 - k*x). - Paul D. Hanna, Sep 05 2012 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 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 a(n) ~ n^n / (2 * (LambertW(n))^n * exp(n+1-n/LambertW(n)) * sqrt(1+LambertW(n))). - Vaclav Kotesovec, Aug 04 2014 MAPLE 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, 1): seq(a(n), n=0..28); # Alois P. Heinz, Jan 15 2018 with(combinat); seq((bell(n) + BellB(n, -1))/2, n = 0..20); # G. C. Greubel, Oct 09 2019 MATHEMATICA 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 *) Table[(BellB[n] + BellB[n, -1])/2, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *) PROG (Sage) def A024430(n) : return add(stirling_number2(n, i) for i in range(0, n+(n+1)%2, 2)) # Peter Luschny, Feb 28 2012 (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 (Magma) a:= func< n | (&+[StirlingSecond(n, 2*k): k in [0..Floor(n/2)]]) >; [a(n): n in [0..25]]; // G. C. Greubel, Oct 09 2019 (GAP) List([0..25], n-> Sum([0..Int(n/2)], k-> Stirling2(n, 2*k)) ); # G. C. Greubel, Oct 09 2019 CROSSREFS Cf. A024429, A121867, A121868, A000110, A000587. Sequence in context: A006219 A353406 A009268 * A357591 A182927 A012408 Adjacent sequences: A024427 A024428 A024429 * A024431 A024432 A024433 KEYWORD nonn AUTHOR Clark Kimberling EXTENSIONS Description changed by N. J. A. Sloane, Jun 14 2003 and again Sep 05 2006 STATUS approved

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Last modified July 12 23:13 EDT 2024. Contains 374257 sequences. (Running on oeis4.)