A003727 Expansion of e.g.f. exp(x * cosh(x)).

1, 1, 1, 4, 13, 36, 181, 848, 3865, 23824, 140521, 871872, 6324517, 44942912, 344747677, 2860930816, 23853473329, 213856723200, 1996865965009, 19099352929280, 193406280000061, 2010469524579328, 21615227339380357, 242177953175506944

Offset: 0

Author

R. H. Hardin

Keywords

• nonn

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..500
• Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function
• Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Crossrefs

Cf. A009233, A191509.

Programs

Magma

m:=50; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x*Cosh(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Sep 09 2018

Mathematica

CoefficientList[Series[E^(x*Cosh[x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 05 2014 *)
Table[Sum[BellY[n, k, Mod[Range[n], 2] Range[n]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

Maxima

a(n):=sum(if n=k then n! else 1/2^k*sum(binomial(n,k)*binomial(k,i)*(k-2*i)^(n-k),i,0,k),k,1,n); /* Vladimir Kruchinin, Aug 22 2010 */

PARI

x='x+O('x^66);
Vec(serlaplace(exp( x * cosh(x) )))
/* Joerg Arndt, Sep 14 2012 */

Formula

a(n) = Sum_{k=1..n} (if n=k then n! otherwise (1/2)^k*Sum_{i=0..k} binomial(n,k)* binomial(k,i)*(k-2*i)^(n-k)), n>0. - Vladimir Kruchinin, Aug 22 2010

a(n) ~ exp(r*cosh(r)-n) * n^n / (r^n * sqrt(3+(r*(r^2-2)*cosh(r))/n)), where r is the root of the equation r*(cosh(r)+r*sinh(r)) = n. - Vaclav Kotesovec, Aug 05 2014

a(n)^(1/n) ~ n*exp(1/(2*LambertW(sqrt(n/2)))-1) / (2*LambertW(sqrt(n/2))). - Vaclav Kotesovec, Aug 05 2014

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * (2*k+1) * a(n-2*k-1). - Ilya Gutkovskiy, Feb 24 2022

Extensions

Extended and formatted by Olivier Gérard, Mar 15 1997