%I #5 May 11 2019 12:21:58
%S 1,1,1,3,14,81,554,4175,33894,292482,2658803,25312031,251337905,
%T 2595476384,27814372541,308814996237,3547597450937,42121414823717,
%U 516406224737906,6531681539263289,85162992707351910,1143744473741844428,15809784290241899546,224756696173450416445,3283701348287927969258,49267186208121297961411,758541179347396245098635,11976195590135148979244826,193765334786286246261399910
%N G.f. A(x) satisfies: Sum_{n>=0} x^n*A(x)^(n^2) = Sum_{n>=0} x^n/(1-n*x)^n.
%e G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 14*x^4 + 81*x^5 + 554*x^6 + 4175*x^7 + 33894*x^8 + 292482*x^9 + 2658803*x^10 + 25312031*x^11 + 251337905*x^12 + ...
%e such that the following series are equal
%e B(x) = 1 + x*A(x) + x^2*A(x)^4 + x^3*A(x)^9 + x^4*A(x)^16 + x^5*A(x)^25 + x^6*A(x)^36 + x^7*A(x)^49 + x^8*A(x)^64 + ...
%e B(x) = 1 + x/(1-x) + x^2/(1-2*x)^2 + x^3/(1-3*x)^3 + x^4/(1-4*x)^4 + x^5/(1-5*x)^5 + x^6/(1-6*x)^6 + x^7/(1-7*x)^7 + x^8/(1-8*x)^8 + ...
%e where
%e B(x) = 1 + x + 2*x^2 + 6*x^3 + 23*x^4 + 104*x^5 + 537*x^6 + 3100*x^7 + 19693*x^8 + 136064*x^9 + 1013345*x^10 + 8076644*x^11 + ... + A080108(n)*x^n + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
%o A[#A] = polcoeff( sum(m=0, #A, x^m*( 1/(1-m*x +x*O(x^#A))^m - Ser(A)^(m^2)) ), #A) ); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A080108.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Apr 30 2019