A182821 Expansion of g.f.: exp( Sum_{n>=1} sigma(5*n)*x^n/n ).

1, 6, 27, 98, 315, 917, 2486, 6345, 15427, 35965, 80897, 176296, 373652, 772381, 1561130, 3091476, 6008896, 11480887, 21591830, 40016045, 73157052, 132052382, 235535752, 415433365, 725043875, 1252857043, 2144601961, 3638413830

Offset: 0

Author

Paul D. Hanna, Dec 05 2010

Keywords

• nonn
• easy

Comments

sigma(5*n) = A000203(5*n), the sum of divisors of 5n.

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 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 917*x^5 + ...
log(A(x)) = 6*x + 18*x^2/2 + 24*x^3/3 + 42*x^4/4 + 31*x^5/5 + 72*x^6/6 + 48*x^7/7 + 90*x^8/8 + ... + sigma(5n)*x^n/n + ...
		

Links

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

Crossrefs

Cf. A000203, A000041; variants: A182818, A182819, A182820.

Cf. Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^k: A035959 (k=1), A160461 (k=2), A277212 (k=5),

this sequence (k=6).

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(5*j))/(1 - x^j)^6: j in [1..(m+2)]]) )); // G. C. Greubel, Nov 18 2018

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

PARI

{a(n)=polcoeff(exp(sum(m=1,n,sigma(5*m)*x^m/m)+x*O(x^n)),n)}

PARI

default(seriesprecision,66); Vec(eta(x^5)/eta(x)^6) \\ Joerg Arndt, Dec 05 2010

PARI

m=30; x='x+O('x^m); Vec(prod(j=1,m+2, (1 - x^(5*j))/(1 - x^j)^6)) \\ G. C. Greubel, Nov 18 2018

Sage

R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(30)
s = prod((1 - x^(5*j))/(1 - x^j)^6 for j in (1..32))
list(s) # G. C. Greubel, Nov 18 2018

Formula

G.f.: A(x) = E(x^5)/E(x)^6 where E(x) = Product_{k>=1} (1-x^k). - Joerg Arndt, Dec 05 2010

a(n) ~ 29^(3/2) * exp(sqrt(58*n/15)*Pi) / (2400*sqrt(3)*n^2). - Vaclav Kotesovec, Nov 28 2016

A(x^5) = P(x)*P(a*x)*P(a^2*x)*P(a^3*x)*P(a^4*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 a = exp(2*Pi*i/5) is a primitive fifth root of unity. - Peter Bala, Jan 24 2017