login
A335766
a(n) is the number of partitions of n into parts congruent to 1, 2, or 4 modulo 6 where only parts congruent to 1 modulo 6 may be repeated.
1
1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 18, 20, 24, 27, 31, 35, 41, 47, 54, 61, 70, 78, 90, 101, 116, 129, 146, 162, 182, 203, 228, 254, 284, 314, 351, 388, 433, 478, 531, 584, 646, 711, 785, 863, 952, 1044, 1149, 1258, 1384, 1513, 1660, 1812, 1983, 2163
OFFSET
0,3
LINKS
J. Lovejoy, Asymmetric generalizations of Schur's theorem, in: Andrews G., Garvan F. (eds) Analytic Number Theory, Modular Forms and q-Hypergeometric Series. ALLADI60 2016. Springer Proceedings in Mathematics & Statistics, vol 221. Springer, Cham.
FORMULA
G.f.: Product_{k>=1} (1+q^(6*k-2))*(1+q^(6*k-4))/(1-q^(6*k-5)).
a(n) ~ Gamma(1/6) * exp(sqrt(2*n)*Pi/3) / (2^(23/12) * sqrt(3) * Pi^(5/6) * n^(7/12)). - Vaclav Kotesovec, Jan 14 2021
EXAMPLE
a(8) = 6, the relevant partitions being [8], [7,1], [4,2,1,1], [4,1,1,1,1],[2,1,1,1,1,1,1], [1,1,1,1,1,1,1,1].
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
b(n-i*j, i-1), j=0..min(n/i, [0, n, 1, 0, 1, 0][irem(i, 6)+1]))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..60); # Alois P. Heinz, Jun 21 2020
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, i - 1], {j, 0, Min[n/i, {0, n, 1, 0, 1, 0}[[Mod[i, 6] + 1]]]}]]];
a[n_] := b[n, n];
a /@ Range[0, 60] (* Jean-François Alcover, Dec 01 2020, after Alois P. Heinz *)
nmax = 60; CoefficientList[Series[Product[(1 + x^(6*k-2)) * (1 + x^(6*k-4)) / (1 - x^(6*k-5)), {k, 1, nmax/6}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 14 2021 *)
CROSSREFS
Cf. A253144.
Sequence in context: A003114 A185227 A217569 * A026823 A025148 A096749
KEYWORD
nonn
AUTHOR
Jeremy Lovejoy, Jun 21 2020
STATUS
approved