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

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

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

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.

Crossrefs

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

Programs

GAP

List([0..25], n-> Sum([0..Int(n/2)], k-> Stirling2(n,2*k)) ); # G. C. Greubel, Oct 09 2019

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

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 *)

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

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

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

Extensions

Description changed by N. J. A. Sloane, Jun 14 2003 and again Sep 05 2006