A241086 -id:A241086

A241089

Number of partitions p of n into distinct parts such that max(p) > 2*(number of parts of p).

+10
8

0, 0, 0, 1, 1, 1, 2, 3, 4, 5, 6, 8, 10, 13, 16, 20, 24, 29, 35, 42, 50, 61, 72, 85, 101, 118, 138, 161, 188, 218, 254, 293, 339, 391, 450, 515, 591, 675, 771, 878, 999, 1135, 1289, 1460, 1652, 1868, 2108, 2376, 2676, 3009, 3379, 3793, 4250, 4760, 5325, 5952

(list; graph; refs; listen; history; text; internal format)

MATHEMATICA

Table[Count[f[n], p_ /; Max[p] <= 2*Length[p]], {n, 0, z}] (* A241086 *)

CROSSREFS

Cf. A241085, A241086, A241087, A241088.

A241087

Number of partitions p of n into distinct parts such that max(p) = 2*(number of parts of p).

+10
7

0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 6, 5, 6, 6, 7, 7, 9, 10, 12, 13, 15, 16, 18, 19, 20, 23, 25, 28, 30, 35, 38, 43, 46, 51, 55, 61, 64, 72, 76, 84, 91, 101, 109, 120, 130, 142, 155, 168, 181, 196, 212, 228, 248, 266, 288, 311, 337

(list; graph; refs; listen; history; text; internal format)

MATHEMATICA

Table[Count[f[n], p_ /; Max[p] <= 2*Length[p]], {n, 0, z}] (* A241086 *)

CROSSREFS

Cf. A241085, A241086, A241088, A241089.

A241085

Number of partitions p of n into distinct parts such that max(p) < 2*(number of parts of p).

+10
5

0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 2, 3, 3, 4, 5, 6, 6, 7, 8, 8, 10, 11, 13, 14, 17, 18, 21, 22, 25, 27, 31, 33, 38, 42, 47, 52, 57, 63, 69, 76, 82, 91, 99, 109, 119, 132, 142, 158, 171, 188, 203, 223, 240, 263, 284, 309, 334, 364, 393, 428, 463, 501, 543, 588

(list; graph; refs; listen; history; text; internal format)

MATHEMATICA

Table[Count[f[n], p_ /; Max[p] <= 2*Length[p]], {n, 0, z}] (* A241086 *)

CROSSREFS

Cf. A241086, A241087, A241088, A241089.

A241088

Number of partitions p of n into distinct parts such that max(p) >= 2*(number of parts of p).

+10
5

0, 0, 1, 1, 1, 2, 3, 4, 4, 6, 7, 10, 12, 15, 18, 22, 26, 32, 39, 46, 56, 66, 78, 91, 108, 125, 147, 171, 200, 231, 269, 309, 357, 410, 470, 538, 616, 703, 801, 913, 1037, 1178, 1335, 1511, 1707, 1929, 2172, 2448, 2752, 3093, 3470, 3894, 4359, 4880, 5455

(list; graph; refs; listen; history; text; internal format)

MATHEMATICA

Table[Count[f[n], p_ /; Max[p] <= 2*Length[p]], {n, 0, z}] (* A241086 *)

CROSSREFS

Cf. A241085, A241086, A241087, A241089.

A241091

Number of partitions p of n into distinct parts such that max(p) <= 1 + 2*(number of parts of p).

+10
3

0, 1, 1, 2, 1, 2, 3, 3, 3, 4, 5, 5, 7, 7, 9, 10, 11, 12, 15, 16, 19, 22, 24, 27, 30, 34, 37, 43, 47, 53, 59, 66, 72, 82, 88, 99, 109, 120, 131, 146, 160, 176, 194, 212, 233, 256, 279, 304, 334, 362, 396, 431, 471, 510, 558, 604, 659, 714, 776, 839, 913, 985

(list; graph; refs; listen; history; text; internal format)

FORMULA

a(n) = A241086(n) + A241092(n) for n >= 0.

MATHEMATICA

Table[Count[f[n], p_ /; Max[p] < 1 + 2*Length[p]], {n, 0, z}] (*A241086*)

CROSSREFS

Cf. A241086, A241092, A241093.

A241092

Number of partitions p of n into distinct parts such that max(p) = 1 + 2*(number of parts of p).

+10
3

0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 7, 7, 8, 9, 10, 10, 12, 13, 15, 17, 19, 21, 25, 26, 29, 32, 35, 38, 42, 46, 51, 57, 62, 69, 76, 83, 90, 100, 107, 117, 127, 139, 150, 165, 178, 195, 212, 231, 250, 273, 294, 319, 346, 373, 402

(list; graph; refs; listen; history; text; internal format)

FORMULA

a(n) + A241086(n) + A241093(n) = A000009(n) for n >= 1.

a(n) = A241091(n) - A241086(n) for n >= 0.

MATHEMATICA

Table[Count[f[n], p_ /; Max[p] < 1 + 2*Length[p]], {n, 0, z}] (*A241086*)

CROSSREFS

Cf. A241086, A241091, A241093, A000009.

A241093

Number of partitions p of n into distinct parts such that max(p) > 1 + 2*(number of parts of p).

+10
3

0, 0, 0, 0, 1, 1, 1, 2, 3, 4, 5, 7, 8, 11, 13, 17, 21, 26, 31, 38, 45, 54, 65, 77, 92, 108, 128, 149, 175, 203, 237, 274, 318, 366, 424, 486, 559, 640, 733, 836, 953, 1084, 1232, 1398, 1583, 1792, 2025, 2286, 2576, 2902, 3262, 3666, 4111, 4610, 5160, 5774

(list; graph; refs; listen; history; text; internal format)

FORMULA

a(n) + A241086(n) + A241093(n) = A000009(n) for n >= 1.

MATHEMATICA

Table[Count[f[n], p_ /; Max[p] < 1 + 2*Length[p]], {n, 0, z}] (*A241086*)

CROSSREFS

Cf. A241086, A241091, A241092, A000009.

A363221

Number of strict integer partitions of n such that (length) * (maximum) <= 2n.

+10
0

1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 15, 19, 23, 26, 29, 37, 39, 49, 55, 62, 71, 84, 93, 108, 118, 141, 149, 188, 193, 217, 257, 279, 318, 369, 376, 441, 495, 572, 587, 692, 760, 811, 960, 1046, 1065, 1307, 1387, 1550, 1703, 1796, 2041, 2295, 2456, 2753, 3014

(list; graph; refs; listen; history; text; internal format)

CROSSREFS

Cf. A111907, A237984, A240219, A241061, A241086, A324521, A324562, A349156, A360068, A360241, A361394, A361852, A361906.