Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #6 Jul 15 2014 12:30:24
%S 1,1,0,3,24,65,480,8827,72576,657729,13754880,215578451,2884992000,
%T 62280478273,1404449120256,27032417472075,640338738708480,
%U 17729894860794497,453894468727209984,12438629293065953059
%N E.g.f. satisfies: A(x) = 1 + x*cosh(x*A(x)).
%H Vaclav Kotesovec, <a href="/A162654/b162654.txt">Table of n, a(n) for n = 0..380</a>
%F a(n) = n!*Sum_{k=0..n} C(n-k+1,k)/(n-k+1) * Sum_{j=0..k} C(k,j)/2^k*(2j-k)^(n-k)/(n-k)!.
%F Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
%F a(n,m) = n!*Sum_{k=0..n} m*C(n-k+m,k)/(n-k+m) * Sum_{j=0..k} C(k,j)/2^k*(2j-k)^(n-k)/(n-k)!.
%F a(n) ~ n^(n-1) * sqrt((2*s-1)/(s-1)) / (exp(n) * r^(n+1)), where r = 0.6184142504137720756... and s = 2.731257206829781545... are roots of the system of equations r^2*sinh(r*s) = 1, s = 1 + r*cosh(r*s). - _Vaclav Kotesovec_, Jul 15 2014
%e E.g.f.: A(x) = 1 + x + 3*x^3/3! + 24*x^4/4! + 65*x^5/5! + 480*x^6/6! +...
%o (PARI) {a(n,m=1)=n!*sum(k=0,n,m*binomial(n-k+m,k)/(n-k+m)*sum(j=0,k,binomial(k,j)/2^k*(2*j-k)^(n-k)/(n-k)!))}
%Y Cf. A162653.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jul 09 2009