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

Author

Ilya Gutkovskiy, Dec 03 2020

Keywords

• nonn

Examples

			a(6) = 4 because we have [6], [3, 2, 1], [2, 2, 2] and [2, 1, 1, 1, 1].
		

Links

• 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

Crossrefs

Cf. A001935, A027193, A042968, A261734, A339404, A339405, A339406.

Programs

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]

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.