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A024429 E.g.f. sinh(exp(x)-1). 13
0, 1, 1, 2, 7, 27, 106, 443, 2045, 10440, 57781, 340375, 2115664, 13847485, 95394573, 690495874, 5235101739, 41428115543, 341177640610, 2917641580783, 25866987547865, 237421321934176, 2252995117706961, 22073206655954547 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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).

REFERENCES

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

A. Fekete and others, Problem 10791, 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.

LINKS

Table of n, a(n) for n=0..23.

Eric Weisstein's World of Mathematics, Stirling Transform.

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)*prod(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)/prod(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

MATHEMATICA

CoefficientList[Series[Sinh[E^x-1], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 04 2014 *)

PROG

(Sage)

def A024429(n) :

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

# Peter Luschny, Feb 28 2012

CROSSREFS

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

Sequence in context: A150591 A150592 A150593 * A136412 A192417 A150594

Adjacent sequences:  A024426 A024427 A024428 * A024430 A024431 A024432

KEYWORD

nonn

AUTHOR

Clark Kimberling

EXTENSIONS

Description changed by N. J. A. Sloane, Sep 05 2006

STATUS

approved

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Last modified December 20 08:42 EST 2014. Contains 252241 sequences.