A325294 G.f. A(x) satisfies: Sum_{n>=0} x^n*A(x)^(n*(n+1)/2) = Sum_{n>=0} x^n/(1-x)^(n^2).

1, 1, 2, 5, 17, 73, 368, 2074, 12663, 82236, 561664, 4004815, 29662508, 227413816, 1800063339, 14681764890, 123207630130, 1062547709801, 9407762681632, 85445941932906, 795514580068247, 7587015660017106, 74078917658328970, 740060483734580171, 7560421405484047766, 78939580213645975075, 841942979579094942598, 9168184497787176646141, 101876790751549107815492

Offset: 0

Author

Paul D. Hanna, Apr 25 2019

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 17*x^4 + 73*x^5 + 368*x^6 + 2074*x^7 + 12663*x^8 + 82236*x^9 + 561664*x^10 + 4004815*x^11 + 29662508*x^12 + ...
such that the following series are equal:
B(x) = 1 + x*A(x) + x^2*A(x)^3 + x^3*A(x)^6 + x^4*A(x)^10 + x^5*A(x)^15 + x^6*A(x)^21 + x^7*A(x)^28 + x^8*A(x)^36 + ...
B(x) = 1 + x/(1-x) + x^2/(1-x)^4 + x^3/(1-x)^9 + x^4/(1-x)^16 + x^5/(1-x)^25 + x^6/(1-x)^36 + x^7/(1-x)^49 + x^8/(1-x)^64 + ...
where
B(x) = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 83*x^5 + 363*x^6 + 1730*x^7 + 8889*x^8 + 48829*x^9 + 284858*x^10 + 1755325*x^11 + ... + A178325(n)*x^n + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A178325, A325289, A326423.

Programs

PARI

{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = -polcoeff( sum(m=0, #A, x^m*( Ser(A)^(m*(m+1)/2) - 1/(1-x +x*O(x^#A))^(m^2)) ), #A) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))