A182818 - OEIS

%I #49 Dec 07 2018 20:37:54

%S 1,3,8,19,41,83,161,299,538,942,1610,2694,4427,7153,11387,17884,27741,

%T 42543,64565,97034,144519,213432,312720,454803,656835,942364,1343596,

%U 1904354,2684008,3762667,5248002,7284132,10063319,13841107,18956002

%N G.f.: exp( Sum_{n>=1} sigma(2n)*x^n/n ).

%C sigma(2n) = A000203(2n), the sum of divisors of 2n (A062731).

%C Compare g.f. to P(x), the g.f. of partition numbers (A000041): P(x) = exp( Sum_{n>=1} sigma(n)*x^n/n ).

%C Number of partitions of n where there are 2 kinds of even parts and 3 kinds of odd parts. - _Ilya Gutkovskiy_, Jan 17 2018

%H Vaclav Kotesovec, <a href="/A182818/b182818.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: A(x) = E(x^2)/E(x)^3 where E(x)=Product_{n>=1} (1 - x^n). - _Joerg Arndt_, Dec 05 2010

%F Conjecture: exp( Sum_{n>=1} sigma(s*n)*x^n/n ) == Product_{d|s} eta(x^d)^(-moebius(d)*sigma(s/d)). - _Joerg Arndt_, Dec 05 2010

%F The ordinary generating function A(x) is the infinite product F(x) * F(x^2) * F(x^3) * ..., where F(x) is the ordinary generating function of A005408. - _Gary W. Adamson_, Jul 15 2012

%F a(n) ~ 5^(3/4) * exp(Pi*sqrt(5*n/3)) / (16 * 3^(3/4) * n^(5/4)). - _Vaclav Kotesovec_, Nov 29 2015

%F From _Peter Bala_, Jan 24 2016: (Start)

%F a(n) = Sum_{k = 0..2*n} (-1)^k*p(k)*p(2*n-k), where p(n) = A000041(n) is the partition function.

%F A(x^2) = 1/Product_{n>=1} (1 - (-x)^n) * 1/Product_{n>=1} (1 - x^n). (End)

%F G.f.: A(x) = Product_{n>=1} (1 - x^(2*n))/(1 - x^n)^3 follows directly from the above formula by Joerg Arndt. - _Paul D. Hanna_, Dec 07 2018

%e G.f.: A(x) = 1 + 3*x + 8*x^2 + 19*x^3 + 41*x^4 + 83*x^5 + 161*x^6 +...

%e log(A(x)) = 3*x + 7*x^2/2 + 12*x^3/3 + 15*x^4/4 + 18*x^5/5 + 28*x^6/6 + 24*x^7/7 + 31*x^8/8 + ... + sigma(2n)*x^n/n + ...

%p with(combinat):

%p seq(add((-1)^k*numbpart(k)*numbpart(2*n - k), k = 0..2*n), n = 0..40);

%t nmax = 40; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2*n])*(x^n/n), {n, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 29 2015 *)

%t nmax = 40; CoefficientList[Series[Product[(1+x^k)/(1-x^k)^2, {k,1,nmax}], {x,0,nmax}], x] (* _Vaclav Kotesovec_, Nov 29 2015 *)

%o (PARI) {a(n)=polcoeff(exp(sum(m=1,n,sigma(2*m)*x^m/m)+x*O(x^n)),n)}

%o (PARI) x='x+O('x^66); Vec(eta(x^2)/eta(x)^3) \\ _Joerg Arndt_, Dec 05 2010

%Y Cf. A062731, A000203, A000041; variants: A182819, A182820, A182821.

%K nonn,easy

%O 0,2

%A _Paul D. Hanna_, Dec 05 2010