A339407 Number of partitions of n into an odd number of parts that are not multiples of 4.

0, 1, 1, 2, 1, 4, 4, 7, 6, 13, 13, 21, 21, 36, 38, 57, 59, 90, 98, 137, 148, 210, 231, 310, 341, 459, 511, 664, 737, 957, 1073, 1357, 1518, 1918, 2156, 2673, 3002, 3712, 4182, 5100, 5737, 6976, 7866, 9460, 10652, 12777, 14402, 17126, 19284, 22867, 25761, 30340, 34139, 40099

Offset

0,4

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) = 4 because we have [6], [3, 2, 1], [2, 2, 2] and [2, 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, 0):
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, A027193, A042968, A261734, A339404, A339405, A339406.

Sequence in context: A289918 A127480 A141446 * A077965 A077967 A296188

Adjacent sequences: A339404 A339405 A339406 * A339408 A339409 A339410

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 03 2020

Status

approved