A290737 Weighted count 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, with respect to a certain weight.

1, 2, 1, 1, 2, -1, 1, 3, -2, 1, 2, 0, 2, 1, 0, -1, 5, 2, -1, 2, -3, 5, 3, -1, 2, 0, 1, 1, 2, -2, 2, 5, 2, -4, 0, 1, -1, 6, 0, 4, -3, -1, 3, -1, 2, 0, 4, -2, 2, 4, -2, 1, 5, -2, -2, -2, 4, 6, 1, 3, -2, 4, -3, -1, -2, 4, 6, 2, 0, -4, 5, 1, 3, -1, 0, 0, 4, -1, -2, 4, -2, 2, 5, 2, 5, -5, -2, 6, -4, 0, -3

Offset: 0

Author

N. J. A. Sloane, Aug 10 2017

Keywords

• sign

Comments

See Andrews (2016) for the definition of the particular weight used here.

Andrews (2016), Theorem 2, shows that A008443(n) = A290735(n) + a(n) + A290739(n).

Links

• George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See Lemma 3.3.

Crossrefs

Cf. A008443, A290733-A290740.

Programs

Maple

M:=201;
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;
D2:=add( q^(2*m+1)*B(q^2,q^4,m)/(1-q^(4*m+2)), m=0..M):
series(D2,q,M); d2seq:=seriestolist(%); BISECT(%,1);

Formula

See Maple code for g.f.