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A260890 The number of overpartitions of n with restricted odd differences.

%I A260890 #26 Nov 15 2024 06:58:09
%S A260890 1,1,3,3,8,9,18,21,39,46,78,93,150,180,276,333,494,597,858,1038,1458,
%T A260890 1764,2424,2931,3960,4783,6360,7671,10068,12123,15720,18894,24249,
%U A260890 29088,36978,44268,55808,66672,83406,99435,123540,146973,181440,215406,264390,313236,382404,452130,549258
%N A260890 The number of overpartitions of n with restricted odd differences.
%C A260890 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 A260890 Alois P. Heinz, <a href="/A260890/b260890.txt">Table of n, a(n) for n = 0..10000</a>
%H A260890 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 A260890 G.f.: Product_{n >= 1} (1-q^(3*n))/((1-q^n)*(1-q^(2*n))).
%F A260890 a(n) ~ sqrt(21) * exp(Pi*sqrt(7*n)/3) / (36*n). - _Vaclav Kotesovec_, Jun 13 2019
%p A260890 with(numtheory):
%p A260890 a:= proc(n) option remember;
%p A260890       `if`(n=0, 1, add(add(d*[1,1,2,0,2,1]
%p A260890       [irem(d, 6)+1], d=divisors(j))*a(n-j), j=1..n)/n)
%p A260890     end:
%p A260890 seq(a(n), n=0..60);  # _Alois P. Heinz_, Aug 08 2015
%t A260890 QP = QPochhammer; QP[q^3]/(QP[q] QP[q^2]) + O[q]^50 // CoefficientList[#, q]& (* _Jean-François Alcover_, Mar 23 2017 *)
%K A260890 nonn
%O A260890 0,3
%A A260890 _Jeremy Lovejoy_, Aug 06 2015