A220513 a(n) = spt(13n+6)/13 where spt(n) = A092269(n).

2, 140, 3042, 38054, 344212, 2488260, 15235620, 81926240, 396603536, 1759312286, 7246532360, 27998586490, 102294344881, 355704104008, 1183463874068, 3784162891544, 11672177600660, 34840196162760, 100912078549712, 284295561826160

Offset: 0

Author

Omar E. Pol, Jan 18 2013

Keywords

• nonn

Comments

That spt(13n+6) == 0 (mod 13) is one of the congruences stated by George E. Andrews. See theorem 2 in the Andrews' paper. See also A220505 and A220507.

Links

• G. E. Andrews, The number of smallest parts in the partitions of n
• G. E. Andrews, F. G. Garvan, and J. Liang, Combinatorial interpretation of congruences for the spt-function
• K. C. Garrett, C. McEachern, T. Frederick, O. Hall-Holt, Fast computation of Andrews' smallest part statistic and conjectured congruences, Discrete Applied Mathematics, 159 (2011), 1377-1380.
• F. G. Garvan, Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank
• F. G. Garvan, Congruences for Andrews' spt-function modulo powers of 5, 7 and 13
• F. G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences, arXiv:1011.1957 [math.NT], 2010.
• K. Ono, Congruences for the Andrews spt-function

Crossrefs

Cf. A076394, A092269, A186113, A220503, A220505, A220507.

Programs

Mathematica

b[n_, i_] := b[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r == 0, q, 0] + Sum[b[n - i*j, i - 1], {j, 0, n/i}]];
spt[n_] := b[n, n];
a[n_] := spt[13 n + 6]/13;
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jan 30 2019, after Alois P. Heinz in A092269 *)

Formula

a(n) = A092269(A186113(n))/13 = A220503(n)/13.