A221141 Third-order spt function.

0, 0, 1, 7, 28, 85, 217, 497, 1036, 2044, 3787, 6797, 11648, 19558, 31703, 50645, 78674, 120932, 181664, 270600, 395682, 574329, 820834, 1166109, 1634668, 2279242, 3142903, 4312063, 5859616, 7927745, 10635129, 14209328, 18846744, 24900807, 32688145, 42761047

Offset

1,4

Links

Jean-François Alcover, Table of n, a(n) for n = 1..50

F. G. Garvan, Higher-order spt functions, preprint.

F. G. Garvan, Higher-order spt functions, arXiv:1008.1207 [math.NT], 2010.

F. G. Garvan, Higher-order spt functions, Adv. Math. 228 (2011), no. 1, 241-265.

Mathematica

om[3, p_List] := Module[{pu, m, f}, pu = Union[p]; m = Length[pu]; f[j_] := Count[p, pu[[j]]]; Binomial[f[1] + 2, 5] + Binomial[f[1] + 1, 3] Sum[ Binomial[f[j] + 1, 2], {j, 2, m}] + f[1] Sum[Binomial[f[j] + 2, 4], {j, 2, m}] + f[1] Sum[Binomial[f[j] + 1, 2] Binomial[f[k] + 1, 2], {j, 2, m}, {k, j + 1, m}]];
spt[3, n_] := Sum[om[3, p], {p, IntegerPartitions[n]}];
Table[spt[3, n], {n, 1, 29}] (* Jean-François Alcover, Mar 30 2020 *)

Crossrefs

Cf. A092269, A221140, A221142, A221143, A221144.

Sequence in context: A369808 A145135 A369807 * A144900 A054469 A369806

Adjacent sequences: A221138 A221139 A221140 * A221142 A221143 A221144

Keyword

nonn

Author

N. J. A. Sloane, Jan 02 2013

Extensions

More terms from Jean-François Alcover, Mar 30 2020

Status

approved