A265254 Number of partitions of n having no even singletons.
1, 1, 1, 2, 3, 4, 6, 8, 11, 15, 19, 25, 34, 43, 54, 70, 89, 111, 140, 174, 216, 268, 328, 402, 495, 601, 727, 883, 1066, 1281, 1540, 1843, 2202, 2627, 3120, 3702, 4392, 5187, 6114, 7206, 8471, 9936, 11644, 13617, 15902, 18548, 21588, 25098, 29156, 33799, 39129
Offset: 0
Keywords
Examples
a(5) = 4 because the partitions [1,1,1,1,1], [1,2,2], [1,1,3], [5] have no even singletons while [1,1,1,2], [2,3], [1,4] do have.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..5000
- Cristina Ballantine and Amanda Welch, Generalizations of POD and PED partitions, arXiv:2308.06136 [math.CO], 2023. See pp. 15-16.
- 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. A265253.
Programs
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Maple
g := mul((1-x^(2*j)+x^(4*j))/(1-x^j), j = 1 .. 80): gser := series(g, x = 0,65): seq(coeff(gser, x, n), n = 0 .. 60); # second Maple program: b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add( `if`(j=1 and i::even, 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) + x^(4*k)) / (1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 01 2016 *) nmax = 50; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(6*k)) / (1 - x^(4*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 01 2016 *)
Formula
G.f.: g(x) = Product_{j>=1} (1 - x^(2j) + x^(4j))/(1-x^j).
a(n) = A265253(n,0).
G.f.: Product_{k>=1} (1 + x^k) * (1 + x^(6*k)) / (1 - x^(4*k)). - Vaclav Kotesovec, Jan 01 2016
a(n) ~ sqrt(5) * exp(sqrt(5*n)*Pi/3) / (3*2^(5/2)*n). - Vaclav Kotesovec, Jan 01 2016