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A024429 Expansion of 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.

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.

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

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

EXAMPLE

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

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 *)

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 May 6 05:30 EDT 2016. Contains 272478 sequences.