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

0,2

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

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

Example

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 + ...

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 *)

Prog

(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
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1-x^j)^8/(1-x^(7*j)): j in [1..m]]) )); // 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

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).

Sequence in context: A161969 A000731 A034433 * A225912 A120081 A173206

Adjacent sequences: A282939 A282940 A282941 * A282943 A282944 A282945

Keyword

sign

Author

Seiichi Manyama, Feb 25 2017

Status

approved