A162649 E.g.f. satisfies: A(x) = exp( x*cosh(x*A(x)) ).

1, 1, 1, 4, 37, 276, 2221, 26888, 397097, 6055696, 103023481, 2047621632, 44856857101, 1051415079872, 26792169643877, 743266588537216, 22085066412427729, 698048232173484288, 23495515539312273265

Offset: 0

Author

Paul D. Hanna, Jul 08 2009

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 37*x^4/4! + 276*x^5/5! +...
log(A(x)) = x*cosh(A(x)) where
cosh(A(x)) = 1 + x^2/2! + 6*x^3/3! + 25*x^4/4! + 160*x^5/5! + 1921*x^6/6! +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Programs

PARI

{a(n,m=1)=sum(k=0,n,binomial(n,k)*m*(n-k+m)^(k-1)*sum(j=0,k,binomial(k,j)*(2*j-k)^(n-k)/2^k))}

Formula

a(n) = Sum_{k=0..n} C(n,k)*(n-k+1)^(k-1)*Sum_{j=0..k} C(k,j)*(2j-k)^(n-k)/2^k.

Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then

a(n,m) = Sum_{k=0..n} C(n,k)*m*(n-k+m)^(k-1)*Sum_{j=0..k} C(k,j)*(2j-k)^(n-k)/2^k.

a(n) ~ sqrt((s*(r*s+sqrt(1+r^4*s^2))) / (r*(1+r*s*sqrt(1+r^4*s^2)))) * n^(n-1) / (exp(n)*r^n), where r = 0.49285491893166753586122556276745..., s = 2.549795671338977846249694869195317... are roots of the system of equations r*cosh(r*s) = log(s), r^2*sinh(r*s) = 1/s. - Vaclav Kotesovec, Jul 14 2014