A339406 Number of partitions of n into an even number of parts that are not multiples of 4.

1, 0, 1, 1, 3, 2, 5, 5, 10, 9, 16, 17, 29, 28, 44, 48, 73, 76, 110, 121, 172, 185, 253, 282, 381, 417, 549, 616, 802, 889, 1137, 1279, 1620, 1810, 2260, 2549, 3161, 3544, 4346, 4906, 5979, 6720, 8120, 9164, 11014, 12392, 14788, 16682, 19820, 22297, 26337, 29682, 34921, 39267

Offset

0,5

Links

Table of n, a(n) for n=0..53.

Cristina Ballantine and Mircea Merca, 4-Regular partitions and the pod function, arXiv:2111.10702 [math.CO], 2021.

Index entries for sequences related to partitions

Formula

G.f.: (1/2) * (Product_{k>=1} (1 - x^(4*k)) / (1 - x^k) + Product_{k>=1} (1 + x^(4*k)) / (1 + x^k)).

a(n) = (A001935(n) + A261734(n)) / 2.

Example

a(6) = 5 because we have [5, 1], [3, 3], [3, 1, 1, 1], [2, 2, 1, 1] and [1, 1, 1, 1, 1, 1].

Maple

b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
      b(n, i-1, t)+`if`(irem(i, 4)=0, 0, b(n-i, min(n-i, i), 1-t))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..55);  # Alois P. Heinz, Dec 03 2020

Mathematica

nmax = 53; CoefficientList[Series[(1/2) (Product[(1 - x^(4 k))/(1 - x^k), {k, 1, nmax}] + Product[(1 + x^(4 k))/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

Crossrefs

Cf. A001935, A027187, A042968, A261734, A339404, A339405, A339407.

Sequence in context: A186545 A008623 A035546 * A182714 A343342 A338470

Adjacent sequences: A339403 A339404 A339405 * A339407 A339408 A339409

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 03 2020

Status

approved