|
%I #27 Jan 30 2019 07:24:18
%S 2,140,3042,38054,344212,2488260,15235620,81926240,396603536,
%T 1759312286,7246532360,27998586490,102294344881,355704104008,
%U 1183463874068,3784162891544,11672177600660,34840196162760,100912078549712,284295561826160
%N a(n) = spt(13n+6)/13 where spt(n) = A092269(n).
%C 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.
%H G. E. Andrews, <a href="http://www.math.psu.edu/vstein/alg/antheory/preprint/andrews/17.pdf">The number of smallest parts in the partitions of n</a>
%H G. E. Andrews, F. G. Garvan, and J. Liang, <a href="http://www.math.psu.edu/andrews/pdf/287.pdf">Combinatorial interpretation of congruences for the spt-function</a>
%H K. C. Garrett, C. McEachern, T. Frederick, O. Hall-Holt, <a href="http://www.deepdyve.com/lp/elsevier/fast-computation-of-andrews-smallest-part-statistic-and-conjectured-pV0SFgLi27/1">Fast computation of Andrews' smallest part statistic and conjectured congruences</a>, Discrete Applied Mathematics, 159 (2011), 1377-1380.
%H F. G. Garvan, <a href="http://www.math.ufl.edu/~fgarvan/papers/spt.pdf">Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank</a>
%H F. G. Garvan, <a href="http://www.math.ufl.edu/~fgarvan/papers/spt2.pdf">Congruences for Andrews' spt-function modulo powers of 5, 7 and 13</a>
%H F. G. Garvan, <a href="http://arxiv.org/abs/1011.1957">Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences</a>, arXiv:1011.1957 [math.NT], 2010.
%H K. Ono, <a href="http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/131.pdf">Congruences for the Andrews spt-function</a>
%F a(n) = A092269(A186113(n))/13 = A220503(n)/13.
%t 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}]];
%t spt[n_] := b[n, n];
%t a[n_] := spt[13 n + 6]/13;
%t Table[a[n], {n, 0, 19}] (* _Jean-François Alcover_, Jan 30 2019, after _Alois P. Heinz_ in A092269 *)
%Y Cf. A076394, A092269, A186113, A220503, A220505, A220507.
%K nonn
%O 0,1
%A _Omar E. Pol_, Jan 18 2013
|
