A220507 - OEIS

%I #26 Jan 30 2019 07:24:11

%S 2,34,260,1498,6956,28024,100953,333680,1026540,2976024,8197962,

%T 21608760,54788100,134217717,318816426,736549424,1659169712,

%U 3652248590,7870890952,16633964444,34522173765,70450341042,141526909340,280158178412

%N a(n) = spt(7n+5)/7 where spt(n) = A092269(n).

%C That spt(7n+5) == 0 (mod 7) is one of the congruences stated by George E. Andrews. See theorem 2 in the Andrews' paper. See also A220505 and A220513.

%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(A017041(n))/7 = A220502(n)/7.

%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[7 n + 5]/7;

%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Jan 30 2019, after _Alois P. Heinz_ in A092269 *)

%Y Cf. A017041, A071746, A092269, A220502, A220505, A220513.

%K nonn

%O 0,1

%A _Omar E. Pol_, Jan 18 2013