This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A003727 M3462 #68 Feb 20 2025 10:29:03
%S A003727 1,1,1,4,13,36,181,848,3865,23824,140521,871872,6324517,44942912,
%T A003727 344747677,2860930816,23853473329,213856723200,1996865965009,
%U A003727 19099352929280,193406280000061,2010469524579328,21615227339380357,242177953175506944
%N A003727 Expansion of e.g.f. exp(x * cosh(x)).
%D A003727 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003727 Vaclav Kotesovec, <a href="/A003727/b003727.txt">Table of n, a(n) for n = 0..500</a>
%H A003727 Vaclav Kotesovec, <a href="http://oeis.org/A216688/a216688.pdf">Asymptotic solution of the equations using the Lambert W-function</a>
%H A003727 Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F A003727 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
%F A003727 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
%F A003727 a(n)^(1/n) ~ n*exp(1/(2*LambertW(sqrt(n/2)))-1) / (2*LambertW(sqrt(n/2))). - _Vaclav Kotesovec_, Aug 05 2014
%F A003727 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
%t A003727 CoefficientList[Series[E^(x*Cosh[x]), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Aug 05 2014 *)
%t A003727 Table[Sum[BellY[n, k, Mod[Range[n], 2] Range[n]], {k, 0, n}], {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)
%o A003727 (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 */
%o A003727 (PARI)
%o A003727 x='x+O('x^66);
%o A003727 Vec(serlaplace(exp( x * cosh(x) )))
%o A003727 /* _Joerg Arndt_, Sep 14 2012 */
%o A003727 (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
%Y A003727 Cf. A009233, A191509.
%K A003727 nonn
%O A003727 0,4
%A A003727 _R. H. Hardin_
%E A003727 Extended and formatted by _Olivier Gérard_, Mar 15 1997