A280454 Expansion of Product_{k>=0} (1 + x^(5*k+1)).

1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 1, 2, 1, 0, 0, 1, 3, 2, 0, 0, 1, 3, 3, 1, 0, 1, 4, 4, 1, 0, 1, 4, 5, 2, 0, 1, 5, 7, 3, 0, 1, 5, 8, 5, 1, 1, 6, 10, 6, 1, 1, 6, 12, 9, 2, 1, 7, 14, 11, 3, 1, 7, 16, 15, 5, 1, 8, 19, 18, 7, 2, 8, 21, 23, 10, 2, 9, 24, 27, 13, 3, 9, 27, 34, 18, 4, 10, 30, 39, 23, 6, 10

Offset

0,18

Comments

Number of partitions of n into distinct parts congruent to 1 mod 5.

Convolution of A281243 and A280454 is A203776. - Vaclav Kotesovec, Jan 18 2017

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

Formula

G.f.: Product_{k>=0} (1 + x^(5*k+1)).

a(n) ~ exp(Pi*sqrt(n)/sqrt(15))/(2*2^(1/5)*15^(1/4)*n^(3/4)) * (1 + (Pi/(240*sqrt(15)) - 3*sqrt(15)/(8*Pi)) / sqrt(n)). - Ilya Gutkovskiy, Jan 03 2017, extended by Vaclav Kotesovec, Jan 24 2017

Example

a(27) = 3 because we have [26, 1], [21, 6] and [11, 16].

Mathematica

nmax = 102; CoefficientList[Series[Product[(1 + x^(5 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, 5] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 18 2017 *)

Crossrefs

Cf. A000700, A016861, A109697, A169975, A261612, A280456, A280457.

Cf. A262928, A147599, A281243, A281244, A281245.

Sequence in context: A284317 A281243 A284314 * A124645 A029409 A335391

Adjacent sequences: A280451 A280452 A280453 * A280455 A280456 A280457

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 03 2017

Status

approved