A282942 Expansion of Product_{k>=1} (1 - q^k)^8/(1 - q^(7*k)) in powers of q.

1, -8, 20, 0, -70, 64, 56, 1, -133, -140, 308, -70, 174, 56, -518, -141, -63, 868, -140, 238, 294, -1029, -1154, -203, 2366, -658, 1296, 350, -1547, -1295, -1666, 3234, -2128, 2534, 2464, -2577, -3087, -609, 5600, -2716, 2435, 294, -3745, -4249, -1015, 8526

Offset: 0

Author

Seiichi Manyama, Feb 25 2017

Keywords

• sign

Examples

			G.f.: 1 - 8*q + 20*q^2 - 70*q^4 + 64*q^5 + 56*q^6 + q^7 - 133*q^8 - 140*q^9 + ...
		

References

• G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 192.

Links

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

Crossrefs

Cf. A282941.

Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), A115110 (k=2), A185654 (k=3), A282937 (k=5), this sequence (k=7).

Programs

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1-x^j)^8/(1-x^(7*j)): j in [1..m]]) )); // G. C. Greubel, Nov 18 2018

Mathematica

nmax = 50; CoefficientList[Series[Product[(1-x^k)^8/(1-x^(7*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* G. C. Greubel, Nov 18 2018 *)

PARI

m=50; x='x+O('x^m); Vec(prod(j=1,m, (1-x^j)^8/(1-x^(7*j)))) \\ G. C. Greubel, Nov 18 2018

Sage

R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(50)
s = prod((1-x^j)^8/(1-x^(7*j)) for j in (1..50))
list(s) # G. C. Greubel, Nov 18 2018

Formula

G.f.: exp( Sum_{n>=1} -sigma(7*n)*q^n/n ). - Seiichi Manyama, Mar 04 2017

a(n) = -(1/n)*Sum_{k=1..n} sigma(7*k)*a(n-k). - Seiichi Manyama, Mar 04 2017