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 1-8 of 8 results.

A339162 Number of compositions (ordered partitions) of n into distinct parts, the least being 1.

Original entry on oeis.org

0, 1, 0, 2, 2, 2, 8, 8, 14, 14, 44, 44, 74, 98, 128, 272, 326, 470, 644, 932, 1106, 2234, 2552, 3800, 4958, 7070, 9068, 12140, 20042, 24674, 34256, 45632, 61814, 80630, 109316, 135572, 217778, 262298, 362744, 466664, 636494, 805454, 1085804, 1375388, 1776938, 2591762
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 25 2020

Keywords

Examples

			a(6) = 8 because we have [5, 1], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 5], [1, 3, 2] and [1, 2, 3].

Links

Crossrefs

Cf. A032020, A096765, A339163, A339164, A339165, A339166.

Programs

  • Maple
    b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-1)*(i+2)/2 `if`(n<1, 0, b(n-1$2, 1)):
seq(a(n), n=0..55);  # Alois P. Heinz, Nov 25 2020
  • Mathematica
    nmax = 45; CoefficientList[Series[Sum[k! x^(k (k + 1)/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 + 1)/2) / Product_{j=1..k-1} (1 - x^j).

A339163 Number of compositions (ordered partitions) of n into distinct parts, the least being 2.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 2, 2, 2, 8, 8, 14, 14, 20, 44, 50, 74, 104, 128, 158, 326, 356, 524, 698, 986, 1160, 1592, 2606, 3158, 4316, 5708, 7706, 10082, 12920, 16136, 25718, 30614, 41756, 53396, 71978, 91058, 122144, 149384, 193670, 279614, 342860, 447764, 581234
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 25 2020

Keywords

Examples

			a(9) = 8 because we have [7, 2], [4, 3, 2], [4, 2, 3], [3, 4, 2], [3, 2, 4], [2, 7], [2, 4, 3] and [2, 3, 4].

Links

Crossrefs

Cf. A002865, A032022, A096749, A339162, A339164, A339165, A339166.

Programs

  • Maple
    b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-2)*(i+3)/2 `if`(n<2, 0, b(n-2$2, 1)):
seq(a(n), n=0..55);  # Alois P. Heinz, Nov 25 2020
  • Mathematica
    nmax = 47; CoefficientList[Series[Sum[k! x^(k (k + 3)/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 + 3)/2) / Product_{j=1..k-1} (1 - x^j).

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).

A339165 Number of compositions (ordered partitions) of n into distinct parts, the least being 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 8, 8, 14, 14, 20, 20, 26, 50, 56, 80, 110, 134, 164, 212, 242, 410, 464, 632, 806, 1118, 1292, 1724, 2042, 2594, 3752, 4448, 5726, 7382, 9524, 12020, 15122, 18602, 23264, 28424, 39830, 46670, 60476, 74780
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 25 2020

Keywords

Examples

			a(15) = 8 because we have [11, 4], [6, 5, 4], [6, 4, 5], [5, 6, 4], [5, 4, 6], [4, 11], [4, 6, 5] and [4, 5, 6].

Links

Crossrefs

Cf. A026797, A026825, A339102, A339162, A339163, A339164, A339166.

Programs

  • Maple
    b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-4)*(i+5)/2 `if`(n<4, 0, b(n-4$2, 1)):
seq(a(n), n=0..55);  # Alois P. Heinz, Nov 25 2020
  • Mathematica
    nmax = 52; CoefficientList[Series[Sum[k! x^(k (k + 7)/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 + 7)/2) / Product_{j=1..k-1} (1 - x^j).

A339169 Number of compositions (ordered partitions) of n into distinct parts, the least being 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 8, 8, 14, 14, 20, 20, 26, 26, 32, 56, 62, 86, 116, 140, 170, 218, 248, 296, 350, 518, 572, 764, 938, 1250, 1448, 1880, 2198, 2774, 3212, 3908, 5210, 6146, 7568, 9368, 11750, 14510, 17756, 21476, 26402, 31826, 38432, 45536
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 25 2020

Keywords

Examples

			a(21) = 8 because we have [15, 6], [8, 7, 6], [8, 6, 7], [7, 8, 6], [7, 6, 8], [6, 15], [6, 8, 7] and [6, 7, 8].

Links

Crossrefs

Cf. A026799, A026827, A339104, A339162, A339163, A339164, A339165, A339166, A339170, A339171, A339172.

Programs

  • Mathematica
    nmax = 62; CoefficientList[Series[Sum[k! x^(k (k + 11)/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 + 11)/2) / Product_{j=1..k-1} (1 - x^j).

A339170 Number of compositions (ordered partitions) of n into distinct parts, the least being 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 8, 8, 14, 14, 20, 20, 26, 26, 32, 32, 62, 62, 92, 116, 146, 170, 224, 248, 302, 350, 404, 572, 650, 818, 1016, 1328, 1526, 1958, 2300, 2852, 3314, 4010, 4592, 6248, 6974, 8750, 10436, 13196, 15722, 19442, 22952
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 25 2020

Keywords

Examples

			a(24) = 8 because we have [17, 7], [9, 8, 7], [9, 7, 8], [8, 9, 7], [8, 7, 9], [7, 17], [7, 9, 8] and [7, 8, 9].

Links

Crossrefs

Cf. A026800, A026828, A339108, A339162, A339163, A339164, A339165, A339166, A339169, A339171, A339172.

Programs

  • Mathematica
    nmax = 64; CoefficientList[Series[Sum[k! x^(k (k + 13)/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 + 13)/2) / Product_{j=1..k-1} (1 - x^j).

A339171 Number of compositions (ordered partitions) of n into distinct parts, the least being 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 8, 8, 14, 14, 20, 20, 26, 26, 32, 32, 38, 62, 68, 92, 122, 146, 176, 224, 254, 302, 356, 404, 458, 650, 704, 896, 1094, 1406, 1604, 2060, 2378, 2954, 3416, 4112, 4694, 5654, 7076, 8156, 9842
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 25 2020

Keywords

Examples

			a(27) = 8 because we have [19, 8], [10, 9, 8], [10, 8, 9], [9, 10, 8], [9, 8, 10], [8, 19], [8, 10, 9] and [8, 9, 10].

Links

Crossrefs

Cf. A026801, A026829, A339109, A339162, A339163, A339164, A339165, A339166, A339169, A339170, A339172.

Programs

  • Mathematica
    nmax = 65; CoefficientList[Series[Sum[k! x^(k (k + 15)/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 + 15)/2) / Product_{j=1..k-1} (1 - x^j).

A339172 Number of compositions (ordered partitions) of n into distinct parts, the least being 9.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 8, 8, 14, 14, 20, 20, 26, 26, 32, 32, 38, 38, 68, 68, 98, 122, 152, 176, 230, 254, 308, 356, 410, 458, 536, 704, 782, 974, 1172, 1484, 1706, 2138, 2480, 3056, 3518, 4214, 4820, 5756
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 25 2020

Keywords

Examples

			a(30) = 8 because we have [21, 9], [11, 10, 9], [11, 9, 10], [10, 11, 9], [10, 9, 11], [9, 21], [9, 11, 10] and [9, 10, 11].

Links

Crossrefs

Cf. A026802, A026830, A339110, A339162, A339163, A339164, A339165, A339166, A339169, A339170, A339171.

Programs

  • Mathematica
    nmax = 67; CoefficientList[Series[Sum[k! x^(k (k + 17)/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 + 17)/2) / Product_{j=1..k-1} (1 - x^j).
Showing 1-8 of 8 results.

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