A003727 Expansion of e.g.f. exp(x * cosh(x)). (Formerly M3462)

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

0,4

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.

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

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 *)

Prog

(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 */
(Magma) m:=50; R<x>:=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

Crossrefs

Cf. A009233, A191509.

Sequence in context: A067635 A222425 A222189 * A103082 A279111 A299111

Adjacent sequences: A003724 A003725 A003726 * A003728 A003729 A003730

Keyword

nonn

Author

R. H. Hardin

Extensions

Extended and formatted by Olivier Gérard, Mar 15 1997

Status

approved