A203013 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (2*A(x)^n - 1)^n * x^n/n!.

1, 1, 5, 55, 993, 24871, 802873, 31793035, 1493163745, 81186783535, 5018214016041, 347636382949747, 26685235607680081, 2248760378885064487, 206430769607981879353, 20507793044444903462251, 2192507508237447321800385, 251034864831917236610746207

Offset: 0

Author

Paul D. Hanna, Dec 27 2011

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 993*x^4/4! + 24871*x^5/5! +...
where the e.g.f. satisfies following series identity:
A(x) = 1 + (2*A(x)-1)*x + (2*A(x)^2-1)^2*x^2/2! + (2*A(x)^3-1)^3*x^3/3! + (2*A(x)^4-1)^4*x^4/4! +...
is equal to
A(x) = exp(-x) + 2*A(x)*exp(-x*A(x))*x + 2^2*A(x)^4*exp(-x*A(x)^2)*x^2/2! + 2^3*A(x)^9*exp(-x*A(x)^3)*x^3/3! + 2^4*A(x)^16*exp(-x*A(x)^4)*x^4/4! +...
		

Crossrefs

Cf. A202999, A361053, A361054, A361055, A361056, A361057.

Cf. A361540.

Programs

PARI

{a(n)=local(A=1+x);for(i=1,n,A=sum(k=0, n, (2*A^k-1+x*O(x^n))^k*x^k/k!));n!*polcoeff(A, n)}

PARI

{a(n)=local(A=1+x);for(i=1,n,A=sum(k=0, n, 2^k*A^(k^2)*exp(-A^k*x+x*O(x^n))*x^k/k!));n!*polcoeff(A, n)}

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following.

(1) A(x) = Sum_{n>=0} (2*A(x)^n - 1)^n * x^n/n!.

(2) A(x) = Sum_{n>=0} 2^n * A(x)^(n^2) * exp(-x*A(x)^n) * x^n/n!.

a(n) = Sum_{k=0..n} A361540(n,k) * 2^(n-k) * (-1)^k. - Paul D. Hanna, Mar 20 2023