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A356507
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G.f.: Sum_{n>=0} x^(n*(n+1)/2) * P(x)^n, where P(x) is the partition function (A000041).
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1
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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
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OFFSET
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0,4
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LINKS
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FORMULA
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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.
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EXAMPLE
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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 + ...
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PROG
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(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), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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