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.

A212543 Number of partitions of n containing at least one part m-3 if m is the largest part.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 8, 10, 17, 22, 33, 42, 60, 75, 103, 128, 169, 209, 271, 331, 421, 513, 642, 777, 963, 1158, 1421, 1703, 2070, 2471, 2985, 3546, 4257, 5043, 6019, 7105, 8443, 9933, 11752, 13790, 16247, 19012, 22326, 26052, 30492, 35500, 41420, 48108, 55980
Offset: 3

Views

Author

Alois P. Heinz, May 20 2012

Keywords

Examples

			a(5) = 1: [4,1].
a(6) = 1: [4,1,1].
a(7) = 3: [4,1,1,1], [4,2,1], [5,2].
a(8) = 4: [4,1,1,1,1], [4,2,1,1], [4,3,1], [5,2,1].
a(9) = 8: [4,1,1,1,1,1], [4,2,1,1,1], [4,2,2,1], [4,3,1,1], [4,4,1], [5,2,1,1], [5,2,2], [6,3].

Links

Crossrefs

Column k=3 of A212551.

Programs

  • Maple
    b:= proc(n, i) option remember;
      `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
    end:
a:= n-> add(b(n-2*m-3, min(n-2*m-3, m+3)), m=1..(n-3)/2):
seq(a(n), n=3..60);
  • Mathematica
    Table[Count[IntegerPartitions[n],?(MemberQ[#,#[[1]]-3]&)],{n,3,60}] (* _Harvey P. Dale, Mar 01 2015 *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + If[i > n, 0, b[n - i, i]]];
a[n_] := Sum[b[n - 2 m - 3, Min[n - 2 m - 3, m + 3]], {m, 1, (n - 3)/2}];
a /@ Range[3, 60] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

G.f.: Sum_{i>0} x^(2*i+3) / Product_{j=1..3+i} (1-x^j).

A212544 Number of partitions of n containing at least one part m-4 if m is the largest part.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 8, 11, 18, 24, 37, 48, 69, 89, 122, 155, 207, 259, 337, 419, 534, 657, 827, 1008, 1252, 1518, 1864, 2246, 2736, 3276, 3960, 4722, 5668, 6727, 8032, 9492, 11274, 13279, 15696, 18424, 21694, 25380, 29772, 34736, 40604, 47244, 55060, 63897
Offset: 4

Views

Author

Alois P. Heinz, May 20 2012

Keywords

Examples

			a(6) = 1: [5,1].
a(7) = 1: [5,1,1].
a(8) = 3: [5,1,1,1], [5,2,1], [6,2].
a(9) = 4: [5,1,1,1,1], [5,2,1,1], [5,3,1], [6,2,1].
a(10) = 8: [5,1,1,1,1,1], [5,2,1,1,1], [5,2,2,1], [5,3,1,1], [5,4,1], [6,2,1,1], [6,2,2], [7,3].

Links

Crossrefs

Column k=4 of A212551.

Programs

  • Maple
    b:= proc(n, i) option remember;
      `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
    end:
a:= n-> add(b(n-2*m-4, min(n-2*m-4, m+4)), m=1..(n-4)/2):
seq(a(n), n=4..60);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + If[i > n, 0, b[n - i, i]]];
a[n_] := Sum[b[n - 2 m - 4, Min[n - 2 m - 4, m + 4]], {m, 1, (n - 4)/2}];
a /@ Range[4, 60] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

G.f.: Sum_{i>0} x^(2*i+4) / Product_{j=1..4+i} (1-x^j).

A212545 Number of partitions of n containing at least one part m-5 if m is the largest part.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 8, 11, 19, 25, 39, 52, 75, 98, 137, 175, 236, 300, 393, 493, 635, 787, 997, 1227, 1531, 1869, 2309, 2796, 3420, 4119, 4994, 5979, 7201, 8574, 10260, 12164, 14470, 17082, 20225, 23778, 28025, 32838, 38542, 45011, 52642, 61286, 71434, 82937
Offset: 5

Views

Author

Alois P. Heinz, May 20 2012

Keywords

Examples

			a(7) = 1: [6,1].
a(8) = 1: [6,1,1].
a(9) = 3: [6,1,1,1], [6,2,1], [7,2].
a(10) = 4: [6,1,1,1,1], [6,2,1,1], [6,3,1], [7,2,1].
a(11) = 8: [6,1,1,1,1,1], [6,2,1,1,1], [6,2,2,1], [6,3,1,1], [6,4,1], [7,2,1,1], [7,2,2], [8,3].

Links

Crossrefs

Column k=5 of A212551.

Programs

  • Maple
    b:= proc(n, i) option remember;
      `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
    end:
a:= n-> add(b(n-2*m-5, min(n-2*m-5, m+5)), m=1..(n-5)/2):
seq(a(n), n=5..60);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + If[i > n, 0, b[n - i, i]]];
a[n_] := Sum[b[n - 2 m - 5, Min[n - 2 m - 5, m + 5]], {m, 1, (n - 5)/2}];
a /@ Range[5, 60] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

G.f.: Sum_{i>0} x^(2*i+5) / Product_{j=1..5+i} (1-x^j).

A212546 Number of partitions of n containing at least one part m-6 if m is the largest part.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 8, 11, 19, 26, 40, 54, 79, 104, 146, 190, 257, 330, 436, 552, 715, 896, 1140, 1415, 1777, 2184, 2711, 3308, 4063, 4922, 5995, 7214, 8720, 10435, 12525, 14910, 17793, 21076, 25016, 29507, 34850, 40941, 48148, 56351, 66007, 76995, 89855, 104484
Offset: 6

Views

Author

Alois P. Heinz, May 20 2012

Keywords

Examples

			a(8) = 1: [7,1].
a(9) = 1: [7,1,1].
a(10) = 3: [7,1,1,1], [7,2,1], [8,2].
a(11) = 4: [7,1,1,1,1], [7,2,1,1], [7,3,1], [8,2,1].
a(12) = 8: [7,1,1,1,1,1], [7,2,1,1,1], [7,2,2,1], [7,3,1,1], [7,4,1], [8,2,1,1], [8,2,2], [9,3].

Links

Crossrefs

Column k=6 of A212551.

Programs

  • Maple
    b:= proc(n, i) option remember;
      `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
    end:
a:= n-> add(b(n-2*m-6, min(n-2*m-6, m+6)), m=1..(n-6)/2):
seq(a(n), n=6..60);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + If[i > n, 0, b[n - i, i]]];
a[n_] := Sum[b[n - 2 m - 6, Min[n - 2 m - 6, m + 6]], {m, 1, (n - 6)/2}];
a /@ Range[6, 60] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

G.f.: Sum_{i>0} x^(2*i+6) / Product_{j=1..6+i} (1-x^j).

A212547 Number of partitions of n containing at least one part m-7 if m is the largest part.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 55, 81, 108, 152, 199, 272, 351, 467, 596, 776, 979, 1255, 1566, 1978, 2448, 3054, 3747, 4628, 5635, 6896, 8342, 10125, 12172, 14673, 17537, 21005, 24981, 29748, 35210, 41718, 49161, 57974, 68049, 79902, 93440, 109295
Offset: 7

Views

Author

Alois P. Heinz, May 20 2012

Keywords

Examples

			a(9) = 1: [8,1].
a(10) = 1: [8,1,1].
a(11) = 3: [8,1,1,1], [8,2,1], [9,2].
a(12) = 4: [8,1,1,1,1], [8,2,1,1], [8,3,1], [9,2,1].
a(13) = 8: [8,1,1,1,1,1], [8,2,1,1,1], [8,2,2,1], [8,3,1,1], [8,4,1], [9,2,1,1], [9,2,2], [10,3].

Links

Crossrefs

Column k=7 of A212551.

Programs

  • Maple
    b:= proc(n, i) option remember;
      `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
    end:
a:= n-> add(b(n-2*m-7, min(n-2*m-7, m+7)), m=1..(n-7)/2):
seq(a(n), n=7..60);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + If[i > n, 0, b[n - i, i]]];
a[n_] := Sum[b[n - 2 m - 7, Min[n - 2 m - 7, m + 7]], {m, 1, (n - 7)/2}];
a /@ Range[7, 60] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

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

A212548 Number of partitions of n containing at least one part m-8 if m is the largest part.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 56, 82, 110, 156, 205, 281, 366, 488, 627, 821, 1041, 1340, 1684, 2135, 2657, 3331, 4108, 5095, 6238, 7663, 9315, 11354, 13709, 16588, 19915, 23936, 28580, 34154, 40573, 48225, 57031, 67452, 79428, 93530, 109695, 128639
Offset: 8

Views

Author

Alois P. Heinz, May 20 2012

Keywords

Examples

			a(10) = 1: [9,1].
a(11) = 1: [9,1,1].
a(12) = 3: [9,1,1,1], [9,2,1], [10,2].
a(13) = 4: [9,1,1,1,1], [9,2,1,1], [9,3,1], [10,2,1].
a(14) = 8: [9,1,1,1,1,1], [9,2,1,1,1], [9,2,2,1], [9,3,1,1], [9,4,1], [10,2,1,1], [10,2,2], [11,3].

Links

Crossrefs

Column k=8 of A212551.

Programs

  • Maple
    b:= proc(n, i) option remember;
      `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
    end:
a:= n-> add(b(n-2*m-8, min(n-2*m-8, m+8)), m=1..(n-8)/2):
seq(a(n), n=8..60);
  • Mathematica
    Table[Count[IntegerPartitions[n],?(MemberQ[#,Max[#]-8]&)],{n,8,55}] (* _Harvey P. Dale, May 05 2016 *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + If[i > n, 0, b[n - i, i]]];
a[n_] := Sum[b[n - 2 m - 8, Min[n - 2 m - 8, m + 8]], {m, 1, (n - 8)/2}];
a /@ Range[8, 60] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

G.f.: Sum_{i>0} x^(2*i+8) / Product_{j=1..8+i} (1-x^j).

A212549 Number of partitions of n containing at least one part m-9 if m is the largest part.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 56, 83, 111, 158, 209, 287, 375, 503, 648, 852, 1086, 1403, 1770, 2255, 2817, 3546, 4393, 5469, 6723, 8294, 10120, 12382, 15011, 18228, 21965, 26497, 31749, 38069, 45383, 54114, 64204, 76176, 89975, 106259, 124998, 146987
Offset: 9

Views

Author

Alois P. Heinz, May 20 2012

Keywords

Examples

			a(11) = 1: [10,1].
a(12) = 1: [10,1,1].
a(13) = 3: [10,1,1,1], [10,2,1], [11,2].
a(14) = 4: [10,1,1,1,1], [10,2,1,1], [10,3,1], [11,2,1].
a(15) = 8: [10,1,1,1,1,1], [10,2,1,1,1], [10,2,2,1], [10,3,1,1], [10,4,1], [11,2,1,1], [11,2,2], [12,3].

Links

Crossrefs

Column k=9 of A212551.

Programs

  • Maple
    b:= proc(n, i) option remember;
      `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
    end:
a:= n-> add(b(n-2*m-9, min(n-2*m-9, m+9)), m=1..(n-9)/2):
seq(a(n), n=9..60);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + If[i > n, 0, b[n - i, i]]];
a[n_] := Sum[b[n - 2 m - 9, Min[n - 2 m - 9, m + 9]], {m, 1, (n - 9)/2}];
a /@ Range[9, 60] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n],?(MemberQ[#,#[[1]]-9]&)],{n,9,60}] (* _Harvey P. Dale, Jun 08 2022 *)

Formula

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

A212550 Number of partitions of n containing at least one part m-10 if m is the largest part.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 56, 83, 112, 159, 211, 291, 381, 512, 663, 873, 1117, 1448, 1833, 2342, 2938, 3708, 4611, 5760, 7105, 8792, 10769, 13215, 16077, 19585, 23679, 28651, 34447, 41424, 49541, 59248, 70509, 83892, 99390, 117695, 138846, 163708
Offset: 10

Views

Author

Alois P. Heinz, May 20 2012

Keywords

Examples

			a(12) = 1: [11,1].
a(13) = 1: [11,1,1].
a(14) = 3: [11,1,1,1], [11,2,1], [12,2].
a(15) = 4: [11,1,1,1,1], [11,2,1,1], [11,3,1], [12,2,1].
a(16) = 8: [11,1,1,1,1,1], [11,2,1,1,1], [11,2,2,1], [11,3,1,1], [11,4,1], [12,2,1,1], [12,2,2], [13,3].

Links

Crossrefs

Column k=10 of A212551.

Programs

  • Maple
    b:= proc(n, i) option remember;
      `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
    end:
a:= n-> add(b(n-2*m-10, min(n-2*m-10, m+10)), m=1..(n-10)/2):
seq(a(n), n=10..60);
  • Mathematica
    Table[Count[IntegerPartitions[n],?(MemberQ[#,#[[1]]-10]&)],{n,10,60}] (* _Harvey P. Dale, Feb 10 2015 *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + If[i > n, 0, b[n - i, i]]];
a[n_] := Sum[b[n - 2m - 10, Min[n - 2m - 10, m + 10]], {m, 1, (n - 10)/2}];
a /@ Range[10, 60] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

G.f.: Sum_{i>0} x^(2*i+10) / Product_{j=1..10+i} (1-x^j).
Showing 1-8 of 8 results.

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