%I #8 Aug 14 2022 15:29:56
%S 1,1,1,3,5,10,18,34,60,109,192,339,591,1027,1768,3032,5165,8755,14766,
%T 24786,41417,68912,114193,188478,309939,507821,829197,1349437,2189105,
%U 3540253,5708422,9177939,14715345,23530180,37527544,59700283,94741244,149991677
%N G.f.: Sum_{n>=0} x^(n*(n+1)/2) * P(x)^n, where P(x) is the partition function (A000041).
%H Paul D. Hanna, <a href="/A356507/b356507.txt">Table of n, a(n) for n = 0..820</a>
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n equals the following expressions involving P(x), the partition function (A000041).
%F (1) A(x) = Sum_{n>=0} x^(n*(n+1)/2) * P(x)^n.
%F (2) A(x) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k>=1} (1 - x^k)^n.
%F (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)).
%F (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.
%e 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 + ...
%e such that
%e 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 + ...
%e where P(x) is the partition function and begins
%e 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 + ...
%o (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)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A000041, A008485.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Aug 11 2022