A182818 G.f.: exp( Sum_{n>=1} sigma(2n)*x^n/n ).

1, 3, 8, 19, 41, 83, 161, 299, 538, 942, 1610, 2694, 4427, 7153, 11387, 17884, 27741, 42543, 64565, 97034, 144519, 213432, 312720, 454803, 656835, 942364, 1343596, 1904354, 2684008, 3762667, 5248002, 7284132, 10063319, 13841107, 18956002

Offset: 0

Author

Paul D. Hanna, Dec 05 2010

Keywords

• nonn
• easy

Comments

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

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

Number of partitions of n where there are 2 kinds of even parts and 3 kinds of odd parts. - Ilya Gutkovskiy, Jan 17 2018

Examples

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

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

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

Programs

Maple

with(combinat):
seq(add((-1)^k*numbpart(k)*numbpart(2*n - k), k = 0..2*n), n = 0..40);

Mathematica

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

PARI

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

PARI

x='x+O('x^66); Vec(eta(x^2)/eta(x)^3) \\ Joerg Arndt, Dec 05 2010

Formula

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

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

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

a(n) ~ 5^(3/4) * exp(Pi*sqrt(5*n/3)) / (16 * 3^(3/4) * n^(5/4)). - Vaclav Kotesovec, Nov 29 2015

From Peter Bala, Jan 24 2016: (Start)

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

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

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