A182819 G.f.: exp( Sum_{n>=1} sigma(3n)*x^n/n ).

1, 4, 14, 39, 101, 238, 533, 1131, 2314, 4566, 8763, 16376, 29939, 53612, 94302, 163112, 277953, 467064, 774943, 1270528, 2060331, 3306771, 5256579, 8280649, 12934125, 20040761, 30817437, 47048638, 71339593, 107469716, 160898163

Offset: 0

Author

Paul D. Hanna, Dec 05 2010

Keywords

• nonn
• easy

Comments

sigma(3n) = A000203(3n), the sum of divisors of 3n (A144613).

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

In general, if r>0 and g.f. = Product_{k>=1} (1 - x^(r*k))/(1 - x^k)^(r+1) then a(n) ~ (r+1-1/r)^((r+1)/4) * exp(Pi*sqrt(2*(r+1-1/r)*n/3)) / (sqrt(r) * 2^((3*r+5)/4) * 3^((r+1)/4) * n^((r+3)/4)). - Vaclav Kotesovec, Nov 28 2016

Examples

			G.f.: A(x) = 1 + 4*x + 14*x^2 + 39*x^3 + 101*x^4 + 238*x^5 +...
log(A(x)) = 4*x + 12*x^2/2 + 13*x^3/3 + 28*x^4/4 + 24*x^5/5 + 39*x^6/6 + 32*x^7/7 + 60*x^8/8 +...+ sigma(3n)*x^n/n +...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A144613, A000203, A000041; variants: A182818, A182820, A182821.

Programs

Maple

w := exp(2*Pi*sqrt(-1)*(1/3)):
with(combinat):
seq(simplify(add(add(w^(j+2*k)*numbpart(j)*numbpart(k)*numbpart(3*n-j-k), k = 0..3*n-j), j = 0..3*n)), n = 0..30);
# Peter Bala, Jan 24 2017

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 26 2016 *)

PARI

{a(n)=polcoeff(exp(sum(m=1,n,sigma(3*m)*x^m/m)+x*O(x^n)),n)}

PARI

default(seriesprecision,66); Vec(eta(x^3)/eta(x)^4)\\ Joerg Arndt, Dec 06 2010

Formula

Generating function A(x) = E(x^3)/E(x)^4 where E(x) = Product_{n>=1} (1-x^n). [Joerg Arndt, Dec 05 2010]

a(n) ~ 11*exp(sqrt(22*n)*Pi/3) / (72*sqrt(6)*n^(3/2)). - Vaclav Kotesovec, Nov 26 2016

From Peter Bala, Jan 24 2016: (Start)

A(x^3) = P(x)*P(w*x)*P(w^2*x), where P(x) = 1/Product_{n>=1} (1 - x^n) is the g.f. for the partition function p(n) = A000041(n), and where w = exp(2*Pi*i/3) is a primitive cube root of unity.

a(n) = Sum_{j = 0..3*n} ( Sum_{k = 0..3*n-j} w^(j+2*k)*p(k)*p(j) *p(3*n-j-k) ). (End)