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

0,4

Links

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

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.

Example

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 + ...

Prog

(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), ", "))

Crossrefs

Cf. A000041, A008485.

Sequence in context: A018165 A054179 A010049 * A094986 A154949 A318248

Adjacent sequences: A356504 A356505 A356506 * A356508 A356509 A356510

Keyword

nonn

Author

Paul D. Hanna, Aug 11 2022

Status

approved