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The number of overpartitions of n into parts congruent to 2, 4, or 5 modulo 6.
1

%I #37 Jan 14 2021 12:01:19

%S 1,0,2,0,4,2,6,4,10,8,18,14,28,24,44,42,68,66,102,104,154,160,226,238,

%T 330,354,476,516,676,742,958,1056,1342,1486,1862,2076,2568,2872,3516,

%U 3940,4782,5370,6464,7268,8686,9774,11606,13070,15428,17380,20408,22986

%N The number of overpartitions of n into parts congruent to 2, 4, or 5 modulo 6.

%H Alois P. Heinz, <a href="/A253136/b253136.txt">Table of n, a(n) for n = 0..10000</a>

%H J. Lovejoy, <a href="http://dx.doi.org/10.1142/S1793042105000157">A theorem on seven-colored overpartitions and its applications</a>, Int. J. Number Theory 1 (2005), 215-224.

%F a(n) ~ Pi^(5/6) * exp(Pi*sqrt(n/2)) / (2^(7/4) * 3^(1/6) * Gamma(1/6) * n^(11/12)). - _Vaclav Kotesovec_, Jan 14 2021

%p series(mul((1+x^(6*k+2))*(1+x^(6*k+4))*(1+x^(6*k+5))/((1-x^(6*k+2))*(1-x^(6*k+4))*(1-x^(6*k+5))), k=0..100), x=0, 100);

%p # second Maple program:

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+

%p `if`(irem(i, 6) in [2, 4, 5], add(2*b(n-i*j, i-1), j=1..n/i), 0)))

%p end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..60); # _Alois P. Heinz_, Jan 04 2019

%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[ MemberQ[{2, 4, 5}, Mod[i, 6]], Sum[2b[n - i j, i-1], {j, 1, n/i}], 0]]];

%t a[n_] := b[n, n];

%t a /@ Range[0, 60] (* _Jean-François Alcover_, Nov 23 2020, after _Alois P. Heinz_ *)

%t nmax = 60; CoefficientList[Series[Product[(1 + x^(6*k+2)) * (1 + x^(6*k+4)) * (1 + x^(6*k+5)) / ((1 - x^(6*k+2)) * (1 - x^(6*k+4)) * (1 - x^(6*k+5))), {k, 0, nmax/6}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 14 2021 *)

%Y Cf. A056970.

%K nonn

%O 0,3

%A _Jeremy Lovejoy_, Mar 23 2015