cp's OEIS Frontend

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

%I A290734 #12 Dec 18 2023 12:07:58
%S A290734 0,-1,2,-2,3,-6,6,-4,6,-9,10,-8,6,-12,14,-6,7,-16,16,-8,10,-18,14,-8,
%T A290734 8,-17,24,-10,6,-24,22,-4,10,-22,20,-16,9,-20,28,-6,8,-32,26,-8,14,
%U A290734 -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
%N A290734 Number of compact partitions of n containing a 1 where each partition is counted with a certain weight.
%C A290734 See Andrews (2016) for the definition of the particular weight used here.
%C A290734 4*A290733(n) + 2*a(n) = (-1)^n*A005875(n) for n > 0.
%H A290734 George E. Andrews, <a href="https://georgeandrews1.github.io/pdf/320.pdf">The Bhargava-Adiga Summation and Partitions</a>, 2016. See Lemma 2.2.
%F A290734 See Maple program for g.f.
%p A290734 M:=101;
%p A290734 B:=proc(a,q,n) local j,t1; global M;
%p A290734 t1:=1;
%p A290734 for j from 0 to M do
%p A290734 t1:=t1*(1-a*q^j)/(1-a*q^(n+j));
%p A290734 od;
%p A290734 t1; end;
%p A290734 # c_1
%p A290734 T1:=add( (-1)^m*q^(m*(m+1)/2)/(B(-q,q,m)*(1+q^m)), m=1..M):
%p A290734 series(T1,q,M); c1seq:=seriestolist(%);
%Y A290734 Cf. A005875, A290733.
%K A290734 sign
%O A290734 0,3
%A A290734 _N. J. A. Sloane_, Aug 10 2017