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

Author

N. J. A. Sloane, Jan 02 2013

Keywords

• nonn

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.

Crossrefs

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

Programs

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 *)

Extensions

More terms from Jean-François Alcover, Mar 30 2020