A290734 Number of compact partitions of n containing a 1 where each partition is counted with a certain weight.

0, -1, 2, -2, 3, -6, 6, -4, 6, -9, 10, -8, 6, -12, 14, -6, 7, -16, 16, -8, 10, -18, 14, -8, 8, -17, 24, -10, 6, -24, 22, -4, 10, -22, 20, -16, 9, -20, 28, -6, 8, -32, 26, -8, 14, -28, 24, -8, 8, -25, 34, -18, 6, -36, 34, -2, 18, -28, 24, -24, 10, -28, 40, -4, 7, -42, 38, -12, 18, -40, 26, -12, 12, -28

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*A290733(n) + 2*a(n) = (-1)^n*A005875(n) for n > 0.

Links

• George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See Lemma 2.2.

Crossrefs

Cf. A005875, A290733.

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_1
T1:=add( (-1)^m*q^(m*(m+1)/2)/(B(-q,q,m)*(1+q^m)), m=1..M):
series(T1,q,M); c1seq:=seriestolist(%);

Formula

See Maple program for g.f.