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

1, 7, 32, 119, 385, 1127, 3057, 7799, 18914, 43950, 98434, 213486, 450051, 925013, 1858355, 3657052, 7062245, 13404195, 25038741, 46086250, 83669927, 149970936, 265608168, 465149039, 806022315, 1382822644, 2350101516, 3958427938

Offset: 0

Author

Paul D. Hanna, Dec 05 2010

Keywords

• nonn
• easy

Comments

sigma(4n) = A000203(4n), the sum of divisors of 4n.

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

Examples

			G.f.: A(x) = 1 + 7*x + 32*x^2 + 119*x^3 + 385*x^4 + 1127*x^5 +...
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 +...
		

Links

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

Crossrefs

Cf. A000203, A000041; variants: A182818, A182819, A182821.

Programs

Maple

with(combinat):
b := n -> add((-1)^k*numbpart(k)*numbpart(n-k), k = 0..n):
seq( add(I^k*b(k)*b(4*n-k), k = 0..4*n), n = 0..30 );
# - Peter Bala, Jan 24 2017

Mathematica

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 *)

PARI

{a(n)=polcoeff(exp(sum(m=1,n,sigma(4*m)*x^m/m)+x*O(x^n)),n)}

PARI

x='x+O('x^66); Vec(eta(x^2)^3/eta(x)^7) \\ Joerg Arndt, Dec 05 2010

Formula

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]

a(n) ~ 11^(5/4) * exp(sqrt(11*n/3)*Pi) / (128 * 3^(5/4) * n^(7/4)). - Vaclav Kotesovec, Nov 28 2016

From Peter Bala, Jan 24 2016: (Start)

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).

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)