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

1,5

Links

Table of n, a(n) for n=1..34.

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[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 *)

Crossrefs

Cf. A092269, A221140, A221141, A221143, A221144.

Sequence in context: A243743 A145458 A145137 * A144902 A128643 A276280

Adjacent sequences: A221139 A221140 A221141 * A221143 A221144 A221145

Keyword

nonn

Author

N. J. A. Sloane, Jan 02 2013

Extensions

More terms from Jinyuan Wang, Aug 08 2021

Status

approved