A356507 G.f.: Sum_{n>=0} x^(n*(n+1)/2) * P(x)^n, where P(x) is the partition function (A000041).

1, 1, 1, 3, 5, 10, 18, 34, 60, 109, 192, 339, 591, 1027, 1768, 3032, 5165, 8755, 14766, 24786, 41417, 68912, 114193, 188478, 309939, 507821, 829197, 1349437, 2189105, 3540253, 5708422, 9177939, 14715345, 23530180, 37527544, 59700283, 94741244, 149991677

Offset: 0

Author

Paul D. Hanna, Aug 11 2022

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 10*x^5 + 18*x^6 + 34*x^7 + 60*x^8 + 109*x^9 + 192*x^10 + 339*x^11 + 591*x^12 + 1027*x^13 + 1768*x^14 + ...
such that
A(x) = 1 + x*P(x) + x^3*P(x)^2 + x^6*P(x)^3 + x^10*P(x)^4 + x^15*P(x)^5 + x^21*P(x)^6 + ... + x^(n*(n+1)/2) * P(x)^n + ...
where P(x) is the partition function and begins
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + 56*x^11 + 77*x^12 + ... + A000041(n)*x^n + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..820

Crossrefs

Cf. A000041, A008485.

Programs

PARI

{a(n) = my(A = sum(m=0,n, x^(m*(m+1)/2) / prod(k=1,n,(1 - x^k +x*O(x^n))^m))); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n equals the following expressions involving P(x), the partition function (A000041).

(1) A(x) = Sum_{n>=0} x^(n*(n+1)/2) * P(x)^n.

(2) A(x) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k>=1} (1 - x^k)^n.

(3) A(x) = Sum_{n>=0} x^n * P(x)^n * Product_{k=1..n} (1 - x^(2*k-1)*P(x))/(1 - x^(2*k)*P(x)).

(4) A(x) = 1/(1 - x*P(x)/(1 + x*(1-x)*P(x)/(1 - x^3*P(x)/(1 + x^2*(1-x^2)*P(x)/(1 - x^5*P(x)/(1 + x^3*(1-x^3)*P(x)/(1 - x^7*P(x)/(1 + x^4*(1-x^4)*P(x)/(1 - ...))))))))), a continued fraction.