A325289 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, 1, 3, 9, 41, 200, 1096, 6440, 40095, 262298, 1790395, 12699751, 93311273, 708519038, 5549751855, 44780255681, 371785828813, 3173019719939, 27813799706468, 250222091088035, 2308676057468240, 21831456961064288, 211449264040904335, 2096345122112307560, 21261235260097878478, 220457711039776064974, 2335722548273384751833

Offset: 0

Author

Paul D. Hanna, Apr 25 2019

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 9*x^4 + 41*x^5 + 200*x^6 + 1096*x^7 + 6440*x^8 + 40095*x^9 + 262298*x^10 + 1790395*x^11 + 12699751*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 + 5*x^3 + 16*x^4 + 57*x^5 + 231*x^6 + 1023*x^7 + 4926*x^8 + 25483*x^9 + 140601*x^10 + ... + A121689(n)*x^n + ...
		

Links

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

Crossrefs

Cf. A121689, A325294, A326424.

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+x +x*O(x^#A))^(m^2)) ),#A) );A[n+1]}
for(n=0,30,print1(a(n),", "))