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

0,2

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

Table of n, a(n) for n=0..79.

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

Formula

See Maple code for g.f.

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 *)

Crossrefs

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

Sequence in context: A205002 A370929 A355029 * A165634 A128282 A146985

Adjacent sequences: A290732 A290733 A290734 * A290736 A290737 A290738

Keyword

nonn

Author

N. J. A. Sloane, Aug 10 2017

Status

approved