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 1-5 of 5 results.

A241416 Number of partitions p of n such that the number of numbers having multiplicity 1 in p is a part and the number of numbers having multiplicity > 1 is not a part.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 2, 1, 2, 3, 4, 4, 8, 14, 16, 28, 33, 47, 61, 83, 98, 131, 157, 201, 248, 312, 379, 480, 589, 730, 903, 1136, 1373, 1725, 2095, 2593, 3129, 3870, 4625, 5677, 6774, 8215, 9759, 11813, 13896, 16738, 19675, 23515, 27580, 32846, 38349, 45528
Offset: 0

Views

Author

Clark Kimberling, Apr 23 2014

Keywords

Examples

			a(6) counts these 2 partitions:  42, 321.

Crossrefs

Cf. A241413, A241414, A241415, A241417, A239737.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] ==  &]]]; e[q_] := Length[DeleteDuplicates[Select[q, Count[q, #] > 1 &]]]
Table[Count[f[n], p_ /; MemberQ[p, u[p]]], {n, 0, z}]  (* A241413 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, e[p]]], {n, 0, z}]  (* A241414 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, e[p]] ], {n, 0, z}] (* A241415 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241416 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241417 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, e[p]] ], {n, 0, z}] (* A239737 *)

Formula

a(n) + A241414(n) + A241415(n) = A239737(n) for n >= 0.

A239737 Number of partitions p of n such that the number of numbers having multiplicity 1 in p is a part or the number of numbers having multiplicity > 1 is a part.

Original entry on oeis.org

0, 1, 1, 2, 2, 5, 7, 10, 13, 21, 28, 38, 54, 77, 99, 137, 180, 236, 306, 398, 504, 644, 807, 1018, 1278, 1599, 1972, 2458, 3039, 3743, 4592, 5659, 6884, 8436, 10235, 12445, 15021, 18204, 21842, 26334, 31501, 37746, 44956, 53707, 63657, 75738, 89536, 106057
Offset: 0

Views

Author

Clark Kimberling, Apr 23 2014

Keywords

Examples

			a(6) counts these 7 partitions:  42, 411, 321, 3111, 2211, 21111, 111111.

Crossrefs

Cf. A241413, A241414, A241415, A241416, A241417, A000041.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] ==  &]]]; e[q_] := Length[DeleteDuplicates[Select[q, Count[q, #] > 1 &]]]
Table[Count[f[n], p_ /; MemberQ[p, u[p]]], {n, 0, z}]  (* A241413 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, e[p]]], {n, 0, z}]  (* A241414 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, e[p]] ], {n, 0, z}] (* A241415 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241416 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241417 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, e[p]] ], {n, 0, z}] (* A239737 *)

Formula

a(n) + A241417(n) = A000041(n) for n >= 0.

A241413 Number of partitions p of n such that the number of numbers having multiplicity 1 in p is a part of p.

Original entry on oeis.org

0, 1, 0, 1, 1, 4, 5, 8, 10, 17, 21, 29, 38, 59, 68, 100, 124, 170, 214, 288, 351, 470, 576, 743, 921, 1176, 1430, 1816, 2214, 2753, 3364, 4176, 5015, 6215, 7478, 9120, 10966, 13351, 15916, 19301, 22982, 27618, 32846, 39354, 46515, 55570, 65598, 77842, 91730
Offset: 0

Views

Author

Clark Kimberling, Apr 23 2014

Keywords

Examples

			a(6) counts these 5 partitions:  42, 411, 321, 3111, 21111; e.g., 411 is counted because 1 part of 411 has multiplicity 1, and 1 is a part of 411.

Crossrefs

Cf. A241414, A241415, A241416, A241417, A239737.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] ==  &]]]; e[q_] := Length[DeleteDuplicates[Select[q, Count[q, #] > 1 &]]]
Table[Count[f[n], p_ /; MemberQ[p, u[p]]], {n, 0, z}]  (* A241413 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, e[p]]], {n, 0, z}]  (* A241414 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, e[p]] ], {n, 0, z}] (* A241415 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241416 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241417 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, e[p]] ], {n, 0, z}] (* A239737 *)

A241415 Number of partitions p of n such that the number of numbers having multiplicity 1 in p is not a part and the number of numbers having multiplicity > 1 is a part.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 3, 4, 7, 9, 16, 18, 31, 37, 56, 66, 92, 110, 153, 174, 231, 275, 357, 423, 542, 642, 825, 990, 1228, 1483, 1869, 2221, 2757, 3325, 4055, 4853, 5926, 7033, 8519, 10128, 12110, 14353, 17142, 20168, 23938, 28215, 33243, 39019, 45968
Offset: 0

Views

Author

Clark Kimberling, Apr 23 2014

Keywords

Examples

			a(6) counts these 2 partitions:  2211, 111111.

Crossrefs

Cf. A241413, A241414, A241416, A241417, A239737, A000041.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] ==  &]]]; e[q_] := Length[DeleteDuplicates[Select[q, Count[q, #] > 1 &]]]
Table[Count[f[n], p_ /; MemberQ[p, u[p]]], {n, 0, z}]  (* A241413 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, e[p]]], {n, 0, z}]  (* A241414 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, e[p]] ], {n, 0, z}] (* A241415 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241416 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241417 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, e[p]] ], {n, 0, z}] (* A239737 *)

Formula

a(n) + A241415(n) + A241416(n) = A239737(n) for n >= 0.

A241417 Number of partitions p of n such that the number of numbers p having multiplicity 1 in p is not a part and the number of numbers having multiplicity > 1 is not a part.

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 4, 5, 9, 9, 14, 18, 23, 24, 36, 39, 51, 61, 79, 92, 123, 148, 195, 237, 297, 359, 464, 552, 679, 822, 1012, 1183, 1465, 1707, 2075, 2438, 2956, 3433, 4173, 4851, 5837, 6837, 8218, 9554, 11518, 13396, 16022, 18697, 22300, 25923, 30873, 35838
Offset: 0

Views

Author

Clark Kimberling, Apr 23 2014

Keywords

Examples

			a(6) counts these 4 partitions:  6, 51, 33, 222.

Crossrefs

Cf. A241413, A241414, A241415, A241416, A239737, A000041.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] ==  &]]]; e[q_] := Length[DeleteDuplicates[Select[q, Count[q, #] > 1 &]]]
Table[Count[f[n], p_ /; MemberQ[p, u[p]]], {n, 0, z}]  (* A241413 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, e[p]]], {n, 0, z}]  (* A241414 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, e[p]] ], {n, 0, z}] (* A241415 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241416 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241417 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, e[p]] ], {n, 0, z}] (* A239737 *)

Formula

a(n) + A239737(n) = A000041(n) for n >= 0.
Showing 1-5 of 5 results.

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