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The number of overpartitions of n with restricted odd differences.
3

%I #26 Nov 15 2024 06:58:09

%S 1,1,3,3,8,9,18,21,39,46,78,93,150,180,276,333,494,597,858,1038,1458,

%T 1764,2424,2931,3960,4783,6360,7671,10068,12123,15720,18894,24249,

%U 29088,36978,44268,55808,66672,83406,99435,123540,146973,181440,215406,264390,313236,382404,452130,549258

%N The number of overpartitions of n with restricted odd differences.

%C The number of overpartitions of n where (i) the difference between successive parts may be odd only if the larger is overlined and (ii) if the smallest part is overlined, then it is odd.

%H Alois P. Heinz, <a href="/A260890/b260890.txt">Table of n, a(n) for n = 0..10000</a>

%H K. Bringmann, J. Dousse, J. Lovejoy, and K. Mahlburg, <a href="https://doi.org/10.37236/5248">Overpartitions with restricted odd differences</a>, Electron. J. Combin. 22 (2015), no.3, paper 3.17.

%F G.f.: Product_{n >= 1} (1-q^(3*n))/((1-q^n)*(1-q^(2*n))).

%F a(n) ~ sqrt(21) * exp(Pi*sqrt(7*n)/3) / (36*n). - _Vaclav Kotesovec_, Jun 13 2019

%p with(numtheory):

%p a:= proc(n) option remember;

%p `if`(n=0, 1, add(add(d*[1,1,2,0,2,1]

%p [irem(d, 6)+1], d=divisors(j))*a(n-j), j=1..n)/n)

%p end:

%p seq(a(n), n=0..60); # _Alois P. Heinz_, Aug 08 2015

%t QP = QPochhammer; QP[q^3]/(QP[q] QP[q^2]) + O[q]^50 // CoefficientList[#, q]& (* _Jean-François Alcover_, Mar 23 2017 *)

%K nonn

%O 0,3

%A _Jeremy Lovejoy_, Aug 06 2015