cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A339164 Number of compositions (ordered partitions) of n into distinct parts, the least being 3.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 2, 8, 8, 14, 14, 20, 20, 50, 50, 80, 104, 134, 158, 212, 356, 410, 578, 752, 1040, 1238, 1646, 1964, 3236, 3674, 5066, 6368, 8720, 10862, 14078, 17180, 22076, 31802, 38378, 49784, 63824, 82670, 104150, 136220, 165980
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 25 2020

Keywords

Examples

			a(12) = 8 because we have [9, 3], [5, 4, 3], [5, 3, 4], [4, 5, 3], [4, 3, 5], [3, 9], [3, 5, 4] and [3, 4, 5].

Links

Crossrefs

Cf. A026796, A026824, A339101, A339162, A339163, A339165, A339166.

Programs

  • Maple
    b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-3)*(i+4)/2 `if`(n<3, 0, b(n-3$2, 1)):
seq(a(n), n=0..55);  # Alois P. Heinz, Nov 25 2020
  • Mathematica
    nmax = 49; CoefficientList[Series[Sum[k! x^(k (k + 5)/2)/Product[1 - x^j, {j, 1, k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=1} k! * x^(k*(k + 5)/2) / Product_{j=1..k-1} (1 - x^j).

A339102 Number of compositions (ordered partitions) of n into distinct parts >= 4.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 15, 15, 23, 29, 37, 43, 57, 87, 101, 137, 175, 235, 279, 363, 431, 665, 757, 1015, 1257, 1683, 2069, 2645, 3199, 4063, 5607, 6759, 8591, 10877, 13837, 17251, 22185, 26871, 33773, 41273, 56047, 66499, 85647, 104811
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 23 2020

Keywords

Examples

			a(9) = 3 because we have [9], [5, 4] and [4, 5].

Links

Crossrefs

Cf. A008484, A017898, A025149, A032020, A032022, A339101, A339103, A339104.

Programs

  • Maple
    b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-3)*(i+4)/2 b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 23 2020
  • Mathematica
    nmax = 52; CoefficientList[Series[Sum[k! x^(k (k + 7)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=0} k! * x^(k*(k + 7)/2) / Product_{j=1..k} (1 - x^j).

A339103 Number of compositions (ordered partitions) of n into distinct parts >= 5.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 9, 15, 17, 23, 31, 37, 45, 57, 65, 101, 115, 151, 189, 255, 293, 383, 451, 565, 777, 921, 1157, 1469, 1855, 2311, 2865, 3495, 4313, 5231, 7063, 8269, 10509, 12849, 16217, 19829, 25171, 30031, 37485, 45183
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 23 2020

Keywords

Examples

			a(11) = 3 because we have [11], [6, 5] and [5, 6].

Links

Crossrefs

Cf. A017899, A025150, A032020, A032022, A185325, A339101, A339102, A339104.

Programs

  • Maple
    b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-4)*(i+5)/2 b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 23 2020
  • Mathematica
    nmax = 54; CoefficientList[Series[Sum[k! x^(k (k + 9)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=0} k! * x^(k*(k + 9)/2) / Product_{j=1..k} (1 - x^j).

A339104 Number of compositions (ordered partitions) of n into distinct parts >= 6.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 17, 17, 25, 31, 39, 45, 59, 65, 79, 115, 129, 165, 209, 269, 313, 403, 471, 585, 683, 941, 1063, 1375, 1641, 2097, 2537, 3161, 3745, 4663, 5535, 6741, 8627, 10241, 12535, 15307, 18849, 22869, 28409
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 23 2020

Keywords

Examples

			a(13) = 3 because we have [13], [7, 6] and [6, 7].

Links

Crossrefs

Cf. A017900, A025151, A032020, A032022, A185326, A339101, A339102, A339103.

Programs

  • Maple
    b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-5)*(i+6)/2 b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 23 2020
  • Mathematica
    nmax = 57; CoefficientList[Series[Sum[k! x^(k (k + 11)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=0} k! * x^(k*(k + 11)/2) / Product_{j=1..k} (1 - x^j).

A339108 Number of compositions (ordered partitions) of n into distinct parts >= 7.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 17, 19, 25, 33, 39, 47, 59, 67, 79, 93, 129, 143, 185, 223, 289, 333, 423, 491, 611, 703, 847, 1089, 1281, 1547, 1889, 2323, 2833, 3417, 4095, 4967, 5939, 7099, 8359, 10653, 12345, 15047, 17993
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 23 2020

Keywords

Examples

			a(15) = 3 because we have [15], [8, 7] and [7, 8].

Links

Crossrefs

Cf. A017901, A025152, A032020, A032022, A185327, A339101, A339102, A339103, A339104, A339109, A339110.

Programs

  • Maple
    b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-6)*(i+7)/2 b(n$2, 0):
seq(a(n), n=0..64);  # Alois P. Heinz, Nov 23 2020
  • Mathematica
    nmax = 60; CoefficientList[Series[Sum[k! x^(k (k + 13)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=0} k! * x^(k*(k + 13)/2) / Product_{j=1..k} (1 - x^j).

A339109 Number of compositions (ordered partitions) of n into distinct parts >= 8.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 19, 19, 27, 33, 41, 47, 61, 67, 81, 93, 107, 143, 163, 199, 243, 309, 353, 443, 517, 631, 729, 873, 995, 1307, 1459, 1795, 2115, 2625, 3089, 3767, 4405, 5371, 6297, 7557, 8771, 10463, 12811, 14911
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 23 2020

Keywords

Examples

			a(17) = 3 because we have [17], [9, 8] and [8, 9].

Links

Crossrefs

Cf. A017902, A025153, A032020, A032022, A185328, A339101, A339102, A339103, A339104, A339108, A339110.

Programs

  • Maple
    b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-7)*(i+8)/2 b(n$2, 0):
seq(a(n), n=0..64);  # Alois P. Heinz, Nov 23 2020
  • Mathematica
    nmax = 64; CoefficientList[Series[Sum[k! x^(k (k + 15)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=0} k! * x^(k*(k + 15)/2) / Product_{j=1..k} (1 - x^j).

A339110 Number of compositions (ordered partitions) of n into distinct parts >= 9.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 19, 21, 27, 35, 41, 49, 61, 69, 81, 95, 107, 121, 163, 177, 219, 263, 329, 373, 469, 537, 657, 755, 899, 1021, 1219, 1485, 1707, 2027, 2417, 2881, 3445, 4077, 4809, 5735, 6755, 7969, 9307
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 23 2020

Keywords

Examples

			a(19) = 3 because we have [19], [10, 9] and [9, 10].

Links

Crossrefs

Cf. A017903, A025154, A032020, A032022, A185329, A339101, A339102, A339103, A339104, A339108, A339109.

Programs

  • Maple
    b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-8)*(i+9)/2 b(n$2, 0):
seq(a(n), n=0..69);  # Alois P. Heinz, Nov 23 2020
  • Mathematica
    nmax = 66; CoefficientList[Series[Sum[k! x^(k (k + 17)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=0} k! * x^(k*(k + 17)/2) / Product_{j=1..k} (1 - x^j).
Showing 1-7 of 7 results.

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