A298689 G.f. A(x) satisfies: A(x) = Sum_{n>=0} binomial( n^2, n) * x^n / A(x)^( n^2 ).

1, 1, 5, 56, 957, 22312, 666666, 24367474, 1051351629, 52144520972, 2915915251326, 181227240764128, 12382862552065170, 922234506009645794, 74345308066436693828, 6449466281781165675666, 599083515375854753327365, 59328642583049975996828036, 6240245388930730524658068558, 694754212357547941002786433000, 81628078642468462576697539116234

Offset: 0

Author

Paul D. Hanna, Jan 24 2018

Keywords

• nonn

Comments

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.

Examples

			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 + ...
such that
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 + ...
more explicitly,
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 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..260

Crossrefs

Cf. A014062, A298690, A298691.

Programs

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]}
for(n=0,30,print1(a(n),", "))

Formula

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