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

0, 1, 1, 2, 7, 27, 106, 443, 2045, 10440, 57781, 340375, 2115664, 13847485, 95394573, 690495874, 5235101739, 41428115543, 341177640610, 2917641580783, 25866987547865, 237421321934176, 2252995117706961, 22073206655954547, 222971522853648704, 2319379362420267753

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

Number of partitions of an n-element set into an odd number of classes. - Peter Luschny, Apr 25 2011

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).

Examples

			G.f. = x + x^2 + 2*x^3 + 7*x^4 + 27*x^5 + 106*x^6 + 443*x^7 + 2045*x^8 + ...
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 226, 4th line of table.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400
• A. Fekete and G. Martin, Problem 10791: Squared Series Yielding Integers, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178.
• S. K. Ghosal, J. K. Mandal, Stirling Transform Based Color Image Authentication, Procedia Technology, 2013 Volume 10, 2013, Pages 95-104.
• Eric Weisstein's World of Mathematics, Stirling Transform.

Crossrefs

Cf. A024430, A121867, A121868, A000110, A000587.

Programs

GAP

List([0..25], n-> Sum([0..Int(n/2)], k-> Stirling2(n,2*k+1)) ); # G. C. Greubel, Oct 09 2019

Magma

a:= func< n | (&+[StirlingSecond(n,2*k+1): 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, 0):
seq(a(n), n=0..28);  # Alois P. Heinz, Jan 15 2018
with(combinat); seq((bell(n) - BellB(n, -1))/2, n = 0..25); # G. C. Greubel, Oct 09 2019

Mathematica

CoefficientList[Series[Sinh[E^x-1], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 04 2014 *)
Table[(BellB[n] - BellB[n, -1])/2, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)

PARI

x='x+O('x^50); concat([0], Vec(serlaplace(sinh(exp(x)-1)))) \\ G. C. Greubel, Nov 12 2017

Sage

def A024429(n) :
    return add(stirling_number2(n,i) for i in range(1,n+n%2,2))
# Peter Luschny, Feb 28 2012

Formula

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.

E.g.f.: sinh(exp(x)-1). - N. J. A. Sloane, Jan 28 2001

a(n) = (A000110(n) - A000587(n)) / 2. - Peter Luschny, Apr 25 2011

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

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

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

G.f.: Sum_{k>=0} x^(2*k+1)/(Product_{i=0..2*k+1} 1-i*x). - Sergei N. Gladkovskii, Jan 06 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, Sep 05 2006