A221142 -id:A221142 - cp's OEIS frontend

A221140 Second-order spt function.

0, 1, 5, 15, 35, 75, 140, 259, 435, 735, 1155, 1841, 2765, 4200, 6125, 8975, 12731, 18179, 25235, 35180, 48055, 65681, 88299, 118895, 157690, 209230, 274510, 359779, 466970, 605740, 778860, 1000462, 1276044, 1624845, 2056355, 2598855, 3265851, 4097763, 5117350

Offset: 1

N. J. A. Sloane, Jan 02 2013

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..60
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.

Cf. A092269, A221141, A221142, A221143, A221144.

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

More terms from Jean-François Alcover, Mar 30 2020

A221141 Third-order spt function.

Original entry on oeis.org

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

N. J. A. Sloane, Jan 02 2013

nonn

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.

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

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 *)

More terms from Jean-François Alcover, Mar 30 2020

A221144 Sixth-order spt function.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 13, 91, 456, 1834, 6293, 19125, 52781, 134643, 321622, 726650, 1564696, 3231635, 6432859, 12395504, 23195905, 42287433, 75274166, 131143033, 223982780, 375713010, 619712403, 1006599177, 1611563058, 2546148075, 3972522337, 6126699846, 9345436668

Offset: 1

N. J. A. Sloane, Jan 02 2013

nonn

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.

Cf. A092269, A221140, A221141, A221142, A221143.

More terms from Jinyuan Wang, Aug 08 2021

A221143 Fifth-order spt function.

Original entry on oeis.org

0, 0, 0, 0, 1, 11, 66, 287, 1013, 3081, 8372, 20824, 48192, 105117, 217945, 433017, 828346, 1534271, 2759132, 4837638, 8283014, 13894554, 22856717, 36968045, 58818578, 92258215, 142699970, 218041302, 329162610, 491683915, 726746512, 1064266795, 1544134041

Offset: 1

N. J. A. Sloane, Jan 02 2013

nonn

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.

Cf. A092269, A221140, A221141, A221142, A221144.

More terms from Jinyuan Wang, Aug 08 2021

A211977 Triangle read by rows, T(n,k) = spt_k(n), in which column k lists the k-th order spt-function (without the first k-1 zeros), n >= 1, k >= 1.

Original entry on oeis.org

1, 3, 1, 5, 5, 1, 10, 15, 7, 1, 14, 35, 28, 9, 1, 26, 75, 85, 45, 11, 1, 35, 140, 217, 166, 66, 13, 1, 57, 259, 497, 505, 287, 91

Offset: 1

Omar E. Pol, Jan 03 2013

The triangle arises from a table due to F. G. Garvan (See the Garvan's paper, page 19).

			Triangle begins:
   1;
   3,   1;
   5,   5,   1;
  10,  15,   7,   1;
  14,  35,  28,   9,   1;
  26,  75,  85,  45,  11,   1;
  35, 140, 217, 166,  66,  13,  1;
  ...

F. G. Garvan, Higher order spt-functions. See page 19.
F. G. Garvan, Higher-order spt functions, Adv. Math. 228 (2011), no. 1, 241-265.

Columns 1-6: A092269, A221140, A221141, A221142, A221143, A221144.

Cf. A000041, A220908, A220909.

cp's OEIS Frontend

A221140 Second-order spt function.

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A221141 Third-order spt function.

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A221144 Sixth-order spt function.

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A221143 Fifth-order spt function.

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A211977 Triangle read by rows, T(n,k) = spt_k(n), in which column k lists the k-th order spt-function (without the first k-1 zeros), n >= 1, k >= 1.

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