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A265254
Number of partitions of n having no even singletons.
4
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
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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 31 2015
STATUS
approved