A265256 Number of partitions of n having no odd singletons (n>=0).

1, 0, 2, 1, 4, 2, 8, 4, 14, 9, 24, 16, 41, 28, 66, 49, 104, 80, 163, 128, 248, 203, 372, 312, 554, 472, 810, 708, 1172, 1042, 1684, 1516, 2390, 2188, 3364, 3118, 4705, 4404, 6522, 6177, 8980, 8584, 12295, 11844, 16718, 16244, 22604, 22120, 30413, 29944, 40692

Offset: 0

Author

Emeric Deutsch, Jan 01 2016

Keywords

• nonn

Examples

			a(5) = 2 because  among the 7 partitions of 5 only [1,1,1,1,1] and [1,1,1,2] have no odd singletons (the others are: [1,2,2], [1,1,3], [2,3], [1,4], [5]).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000
• James A. Sellers, Elementary Proofs of Congruences for POND and PEND Partitions, arXiv:2308.09999 [math.NT], 2023.
• James A. Sellers and Nicolas Allen Smoot, Explaining Unforeseen Congruence Relationships Between PEND and POND Partitions via an Atkin--Lehner Involution, arXiv:2503.16019 [math.NT], 2025

Crossrefs

Cf. A265254, A265255.

Programs

Maple

g := mul((1-x^(2*j-1)+x^(4*j-2))/(1-x^j), j = 1 .. 80): gser := series(g, x = 0, 60): seq(coeff(gser, x, n), n = 0 .. 55);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(j=1 and i::odd, 0, b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 02 2016

Mathematica

nmax = 50; CoefficientList[Series[Product[((1 - x^(2*k-1) + x^(4*k-2))) / (1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 01 2016 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(6*k-3)) / ((1 + x^(2*k-1)) * (1-x^k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 01 2016 *)

Formula

a(n) = A265255(n,0).

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

a(n) ~ sqrt(5) * exp(sqrt(5*n)*Pi/3) / (12*sqrt(2)*n). - Vaclav Kotesovec, Jan 01 2016