A156214 - OEIS

%I #8 Nov 14 2017 03:03:41

%S 1,2,14,256,18734,6932928,11550075900,80606017093632,

%T 2307293302418365718,268696321569450570148864,

%U 126770971088210751226430473604,241680859880056839468193961216049152

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

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

%F 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]

%e G.f.: A(x) = 1 + 2*x + 14*x^2 + 256*x^3 + 18734*x^4 + 6932928*x^5 +...

%e log(A(x)) = 2*x + 24*x^2/2 + 692*x^3/3 + 72704*x^4/4 + 34465932*x^5/5 +...

%e log(A(x)) = 2*xA(x) + 2^4*(xA(x))^2/2 + 2^9*(xA(x))^3/3 + 2^16*(xA(x))^4/4 + ...

%t terms = 12;

%t g[n_] := g[n] = If[n == 0, 1, (1/n)*Sum[2^(k^2)*g[n - k], {k, 1, n}]];

%t G[x_] = Sum[g[n]*x^n, {n, 0, terms}];

%t A[x_] = (1/x)*InverseSeries[Series[x/G[x], {x, 0, terms}], x];

%t CoefficientList[A[x] + O[x]^terms, x] (* _Jean-François Alcover_, Nov 14 2017 *)

%o (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)}

%Y Cf. A000108, A156213.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 06 2009