A221142 Fourth-order spt function.

0, 0, 0, 1, 9, 45, 166, 505, 1341, 3223, 7149, 14916, 29480, 55902, 101892, 180245, 309297, 518859, 849563, 1366441, 2154789, 3348972, 5119981, 7733835, 11520100, 16985374, 24746334, 35735413, 51073008, 72432093, 101794713, 142085314, 196744665, 270764547

Offset: 1

Author

N. J. A. Sloane, Jan 02 2013

Keywords

• nonn

Links

• 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, A221141, A221143, A221144.

Programs

Mathematica

om[4, p_List] := Module[{pu, m, f}, pu = Union[p]; m = Length[pu]; f[j_] := Count[p, pu[[j]]]; Binomial[f[1] + 3, 7] + Binomial[f[1] + 2, 5] Sum[Binomial[f[j] + 1, 2], {j, 2, m}] + Binomial[f[1] + 1, 3] Sum[Binomial[f[j] + 2, 4], {j, 2, m}] + f[1] Sum[Binomial[f[j] + 3, 6], {j, 2, m}] + Binomial[f[1] + 1, 3] Sum[Binomial[f[j] + 1, 2] Binomial[f[k] + 1, 2], {j, 2, m}, {k, j + 1, m}] + f[1] Sum[Binomial[f[j] + 2, 4] Binomial[f[k] + 1, 2], {j, 2, m}, {k, j + 1, m}] + f[1] Sum[Binomial[f[j] + 1, 2] Binomial[f[k] + 2, 4], {j, 2, m}, {k, j + 1, m}] + f[1] Sum[Binomial[f[j] + 1, 2] Binomial[f[k] + 1, 2] Binomial[f[r] + 1, 2], {j, 2, m}, {k, j + 1, m}, {r, k + 1, m}]];
spt[4, n_] := Sum[om[4, p], {p, IntegerPartitions[n]}];
Table[spt[4, n], {n, 1, 35}] (* Jinyuan Wang, Aug 08 2021 *)

Extensions

More terms from Jinyuan Wang, Aug 08 2021