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A182820 G.f.: exp( Sum_{n>=1} sigma(4n)*x^n/n ).

%I A182820 #24 Jan 26 2017 02:45:53
%S A182820 1,7,32,119,385,1127,3057,7799,18914,43950,98434,213486,450051,925013,
%T A182820 1858355,3657052,7062245,13404195,25038741,46086250,83669927,
%U A182820 149970936,265608168,465149039,806022315,1382822644,2350101516,3958427938
%N A182820 G.f.: exp( Sum_{n>=1} sigma(4n)*x^n/n ).
%C A182820 sigma(4n) = A000203(4n), the sum of divisors of 4n.
%C A182820 Compare g.f. to P(x), the g.f. of partition numbers (A000041): P(x) = exp( Sum_{n>=1} sigma(n)*x^n/n ).
%H A182820 Seiichi Manyama, <a href="/A182820/b182820.txt">Table of n, a(n) for n = 0..1000</a>
%F A182820 Generating function A(x) = E(x^2)^3/E(x)^7 where E(x) = Product_{n>=1} (1-x^n). [_Joerg Arndt_, Dec 05 2010]
%F A182820 a(n) ~ 11^(5/4) * exp(sqrt(11*n/3)*Pi) / (128 * 3^(5/4) * n^(7/4)). - _Vaclav Kotesovec_, Nov 28 2016
%F A182820 From _Peter Bala_, Jan 24 2016: (Start)
%F A182820 A(x^4) = P(x)*P(-x)*P(i*x)*P(-i*x), where P(x) = 1/Product_{n>=1} (1 - x^n) is the g.f. for the partition function p(n) = A000041(n).
%F A182820 a(n) = Sum_{k = 0..4*n} i^k*b(k)*b(4*n-k), where b(n) := Sum_{k = 0..n} (-1)^k*p(k)*p(n-k). (End)
%e A182820 G.f.: A(x) = 1 + 7*x + 32*x^2 + 119*x^3 + 385*x^4 + 1127*x^5 +...
%e A182820 log(A(x)) = 7*x + 15*x^2/2 + 28*x^3/3 + 31*x^4/4 + 42*x^5/5 + 60*x^6/6 + 56*x^7/7 + 63*x^8/8 +...+ sigma(4n)*x^n/n +...
%p A182820 with(combinat):
%p A182820 b := n -> add((-1)^k*numbpart(k)*numbpart(n-k), k = 0..n):
%p A182820 seq( add(I^k*b(k)*b(4*n-k), k = 0..4*n), n = 0..30 );
%p A182820 # - _Peter Bala_, Jan 24 2017
%t A182820 nmax = 50; CoefficientList[Series[Product[(1 + x^k)^3/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 28 2016 *)
%o A182820 (PARI) {a(n)=polcoeff(exp(sum(m=1,n,sigma(4*m)*x^m/m)+x*O(x^n)),n)}
%o A182820 (PARI) x='x+O('x^66); Vec(eta(x^2)^3/eta(x)^7) \\ _Joerg Arndt_, Dec 05 2010
%Y A182820 Cf. A000203, A000041; variants: A182818, A182819, A182821.
%K A182820 nonn,easy
%O A182820 0,2
%A A182820 _Paul D. Hanna_, Dec 05 2010