%I #17 Jan 25 2018 15:22:58

%S 1,1,5,56,957,22312,666666,24367474,1051351629,52144520972,

%T 2915915251326,181227240764128,12382862552065170,922234506009645794,

%U 74345308066436693828,6449466281781165675666,599083515375854753327365,59328642583049975996828036,6240245388930730524658068558,694754212357547941002786433000,81628078642468462576697539116234

%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} binomial( n^2, n) * x^n / A(x)^( n^2 ).

%C Compare to: Sum_{n>=0} C(m*n,n) * x^n / (1+x)^(m*n) = (1+x)/(1 - (m-1)*x) holds for fixed m.

%H Paul D. Hanna, <a href="/A298689/b298689.txt">Table of n, a(n) for n = 0..260</a>

%F a(2^k) is odd for k>=0, and a(n) is even elsewhere except at n=0 (conjecture).

%e G.f.: A(x) = 1 + x + 5*x^2 + 56*x^3 + 957*x^4 + 22312*x^5 + 666666*x^6 + 24367474*x^7 + 1051351629*x^8 + 52144520972*x^9 + 2915915251326*x^10 + 181227240764128*x^11 + 12382862552065170*x^12 + ...

%e such that

%e A(x) = 1 + C(1,1)*x/A(x) + C(4,2)*x^2/A(x)^4 + C(9,3)*x^3/A(x)^9 + C(16,4)*x^4/A(x)^16 + C(25,5)*x^5/A(x)^25 + C(36,6)*x^6/A(x)^36 + C(49,7)*x^7/A(x)^49 + ...

%e more explicitly,

%e A(x) = 1 + x/A(x) + 6*x^2/A(x)^4 + 84*x^3/A(x)^9 + 1820*x^4/A(x)^16 + 53130*x^5/A(x)^25 + 1947792*x^6/A(x)^36 + 85900584*x^7/A(x)^49 + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = Vec(sum(m=0,#A,binomial(m^2,m) * x^m/Ser(A)^(m^2) ))); A[n+1]}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A014062, A298690, A298691.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 24 2018