A290733 Number of compact partitions of n where each partition is counted with a certain weight.

0, -1, 2, -1, 0, -3, 3, 2, 0, -3, 1, -2, -1, 0, 5, 3, -2, -4, 1, -2, 1, -3, -1, 4, 2, 1, 6, -3, -3, -6, 1, 2, -2, -1, 2, -4, 3, 4, 4, 3, 2, -8, -1, -2, -1, -4, 0, 4, -2, -1, 4, -3, 3, 0, 7, 1, 3, 2, -6, -6, -5, -4, 4, 2, -2

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.

4*a(n) + 2*A290734(n) = (-1)^n*A005875(n) for n > 0.

Links

• George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See Lemma 2.1.

Crossrefs

Cf. A005875, A290734.

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;
# c_0
t2:=add((-1)^m*q^m*B(-q,q,m-1)/(1+q^m), m=1..M):
series(t2,q,M);
seriestolist(%);

Formula

See Maple program for g.f.