A281245 Expansion of Product_{k>=1} (1 + x^(7*k-1)).

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 2, 3, 1, 0, 0, 0, 0, 3, 4, 1, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 1, 5, 5, 1, 0, 0, 0, 2, 7, 5, 1, 0, 0, 0

Offset: 0

Author

Vaclav Kotesovec, Jan 18 2017

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A262928, A147599, A281243, A281244.

Cf. A261612, A169975, A280454.

Cf. A109708, A281455, A281456, A281457, A281458, A280457, A281459.

Programs

Mathematica

nmax = 200; CoefficientList[Series[Product[(1 + x^(7*k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 7] == 6, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly

Formula

a(n) ~ exp(sqrt(n/21)*Pi) / (2^(13/7)*21^(1/4)*n^(3/4)) * (1 + (13*Pi/(336*sqrt(21)) - 3*sqrt(21)/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Jan 18 2017, extended Jan 24 2017

G.f.: Sum_{k>=0} x^(k*(7*k + 5)/2) / Product_{j=1..k} (1 - x^(7*j)). - Ilya Gutkovskiy, Nov 24 2020