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.

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, 61, 65, 74, 80, 88, 99, 110, 116, 128, 143, 153, 170, 186, 200, 221, 241, 259, 285, 308, 332, 366, 396, 426, 462, 502, 539, 587, 636, 678, 738, 796, 856, 925, 993

Offset: 0

Author

N. J. A. Sloane, Aug 10 2017

Keywords

• nonn

Examples

			For example, the relevant partitions of 7 are 7, 5+1+1, and 1+1+1+1+1+1+1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See (3.4).

Crossrefs

Cf. A008443, A290733, A290734, A290735, A290737, A290738, A290739.

Programs

Maple

M:=101;
B:=proc(a,q,n) local j,t1; global M; t1:=1;
for j from 0 to M do t1:=t1*(1-a*q^j)/(1-a*q^(n+j)); od;
t1; end;
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);

Mathematica

M = 121;
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];
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 *)

Formula

See Maple code for g.f.

a(n) ~ exp(Pi*sqrt(n/6)) / (2^(3/2) * sqrt(n)). - Vaclav Kotesovec, May 24 2018