A280457 Expansion of Product_{k>=0} (1 + x^(7*k+1)).

1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 1, 4, 5, 2, 0, 0, 0, 1, 5, 7, 3, 0, 0, 0, 1, 5, 8, 5, 1, 0, 0, 1, 6, 10, 6, 1, 0, 0, 1, 6, 12, 9, 2, 0, 0, 1, 7, 14, 11, 3, 0, 0, 1, 7, 16, 15, 5

Offset: 0

Author

Ilya Gutkovskiy, Jan 03 2017

Keywords

• nonn

Comments

Number of partitions of n into distinct parts congruent to 1 mod 7.

Examples

			a(37) = 3 because we have [36, 1], [29, 8] and [22, 15].
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015.
• Index entries for related partition-counting sequences

Crossrefs

Cf. A000700, A016993, A169975, A261612, A280454.

Cf. A262928, A147599, A281243, A281244.

Cf. A109703, A281245, A281455, A281456, A281457, A281458, A281459.

Programs

Mathematica

nmax = 105; CoefficientList[Series[Product[(1 + x^(7 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 7] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 18 2017 *)

Formula

G.f.: Product_{k>=0} (1 + x^(7*k+1)).

a(n) ~ exp(Pi*sqrt(n)/sqrt(21))/(2*2^(1/7)*21^(1/4)*n^(3/4)) * (1 + (13*Pi/(336*sqrt(21)) - 3*sqrt(21)/(8*Pi)) / sqrt(n)). - Ilya Gutkovskiy, Jan 03 2017, extended by Vaclav Kotesovec, Jan 24 2017