A162654 - 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