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

0,4

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.

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

Example

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.

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 *)

Crossrefs

Cf. A265253.

Sequence in context: A127419 A262160 A132217 * A303944 A039855 A175868

Adjacent sequences: A265251 A265252 A265253 * A265255 A265256 A265257

Keyword

nonn

Author

Emeric Deutsch, Dec 31 2015

Status

approved