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A290736 Number of partitions of 2n+1 into odd parts in which the largest part appears an odd number of times and all other parts appear twice.

%I A290736 #28 Dec 18 2023 12:11:07
%S A290736 1,2,3,3,4,5,5,7,8,7,10,12,12,15,16,17,21,24,25,28,33,35,41,45,46,56,
%T A290736 61,65,74,80,88,99,110,116,128,143,153,170,186,200,221,241,259,285,
%U A290736 308,332,366,396,426,462,502,539,587,636,678,738,796,856,925,993
%N A290736 Number of partitions of 2n+1 into odd parts in which the largest part appears an odd number of times and all other parts appear twice.
%H A290736 Alois P. Heinz, <a href="/A290736/b290736.txt">Table of n, a(n) for n = 0..10000</a>
%H A290736 George E. Andrews, <a href="https://georgeandrews1.github.io/pdf/320.pdf">The Bhargava-Adiga Summation and Partitions</a>, 2016. See (3.4).
%F A290736 See Maple code for g.f.
%F A290736 a(n) ~ exp(Pi*sqrt(n/6)) / (2^(3/2) * sqrt(n)). - _Vaclav Kotesovec_, May 24 2018
%e A290736 For example, the relevant partitions of 7 are 7, 5+1+1, and 1+1+1+1+1+1+1.
%p A290736 M:=101;
%p A290736 B:=proc(a,q,n) local j,t1; global M; t1:=1;
%p A290736 for j from 0 to M do t1:=t1*(1-a*q^j)/(1-a*q^(n+j)); od;
%p A290736 t1; end;
%p A290736 D34:=add( q^(2*m+1)*B(-q^2,q^4,m)/(1-q^(4*m+2)), m=0..M):series(D34,q,M); d34seq:=seriestolist(%); BISECT(%,1);
%t A290736 M = 121;
%t A290736 B[a_, q_, n_] := Module[{j, t1 = 1}, For[j = 0, j <= M, j++, t1 = t1(1 - a q^j)/(1 - a q^(n+j))]; t1];
%t A290736 seq = Sum[q^(2m+1) B[-q^2, q^4, m]/(1 - q^(4m+2)), {m, 0, M}] + O[q]^M // CoefficientList[#, q]& // Partition[#, 2]& // #[[All, 2]]& (* _Jean-François Alcover_, Dec 16 2020, after Maple *)
%Y A290736 Cf. A008443, A290733, A290734, A290735, A290737, A290738, A290739.
%K A290736 nonn
%O A290736 0,2
%A A290736 _N. J. A. Sloane_, Aug 10 2017