A290735 a(n) = weighted sum over all the self-conjugate partitions of 4n + 1 into odd parts, with respect to a certain weight.

1, 2, 2, 3, 4, 3, 5, 6, 4, 6, 7, 6, 7, 8, 6, 7, 11, 7, 8, 10, 6, 11, 12, 7, 10, 12, 8, 11, 13, 8, 11, 16, 10, 9, 15, 8, 13, 18, 9, 14, 14, 10, 15, 16, 10, 13, 20, 11, 13, 20, 8, 17, 22, 8, 14, 17, 15, 18, 20, 12, 14, 23, 12, 14, 20, 12, 21, 25, 9, 16, 22, 14, 21, 22, 12, 15, 26, 16, 14, 26

Offset: 0

Author

N. J. A. Sloane, Aug 10 2017

Keywords

• nonn

Comments

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

Andrews (2016), Theorem 2, shows that A008443(n) = a(n) + A290737(n) + A290739(n).

Andrews conjectures that a(n) > 0 for all n. The conjecture is known to be true for n <= 1000.

Andrews also conjectures that a(n) > |A290737(n) + A290739(n)| for n >= 2 (see A290740).

Links

• George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See Lemma 3.2.

Crossrefs

Cf. A008443, A290733, A290734, A290736, A290737, A290738, A290739, A290740.

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;
D1:=add( (-1)^m*q^(m*(m+1))/(B(q,q^2,m+1)*(1-q^(2*m+1))), m=0..M):
series(D1,q,M); d1seq:=seriestolist(%);

Mathematica

M = 101;
B[a_, q_, n_] := Module[{j, t1},  t1 = 1; For[j = 0, j <= M, j++, t1 = t1*(1-a*q^j)/(1-a*q^(n+j))]; t1];
D1 = Sum[(-1)^m*q^(m*(m+1))/(B[q, q^2, m+1]*(1-q^(2*m+1))), {m, 0, M}];
Series[D1, {q, 0, M}] // CoefficientList[#, q]& (* Jean-François Alcover, Mar 16 2023, after Maple code *)

Formula

See Maple code for g.f.