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A221142
Fourth-order spt function.
5
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
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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 02 2013
EXTENSIONS
More terms from Jinyuan Wang, Aug 08 2021
STATUS
approved