A156214 G.f.: A(x) = exp( Sum_{n>=1} 2^(n^2)*(x*A(x))^n/n ), a power series in x with integer coefficients.

1, 2, 14, 256, 18734, 6932928, 11550075900, 80606017093632, 2307293302418365718, 268696321569450570148864, 126770971088210751226430473604, 241680859880056839468193961216049152

Offset: 0

Author

Paul D. Hanna, Feb 06 2009

Keywords

• nonn

Comments

Compare to g.f. for Catalan sequence: C(x) = exp( Sum_{n>=1} (x*C(x))^n/n ).

Examples

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 256*x^3 + 18734*x^4 + 6932928*x^5 +...
log(A(x)) = 2*x + 24*x^2/2 + 692*x^3/3 + 72704*x^4/4 + 34465932*x^5/5 +...
log(A(x)) = 2*xA(x) + 2^4*(xA(x))^2/2 + 2^9*(xA(x))^3/3 + 2^16*(xA(x))^4/4 + ...
		

Crossrefs

Cf. A000108, A156213.

Programs

Mathematica

terms = 12;
g[n_] := g[n] = If[n == 0, 1, (1/n)*Sum[2^(k^2)*g[n - k], {k, 1, n}]];
G[x_] = Sum[g[n]*x^n, {n, 0, terms}];
A[x_] = (1/x)*InverseSeries[Series[x/G[x], {x, 0, terms}], x];
CoefficientList[A[x] + O[x]^terms, x] (* Jean-François Alcover, Nov 14 2017 *)

PARI

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

Formula

G.f.: A(x) = (1/x)*Series_Reversion(x/G(x)) where A(x*G(x)) = G(x) is the g.f. of A155200. [Paul D. Hanna, Jun 30 2009]