A336269 -id:A336269 - cp's OEIS frontend

A276429 Number of partitions of n containing no part i of multiplicity i.

1, 0, 2, 2, 3, 5, 8, 9, 16, 19, 29, 36, 53, 65, 92, 115, 154, 195, 257, 318, 419, 516, 663, 821, 1039, 1277, 1606, 1963, 2441, 2978, 3675, 4454, 5469, 6603, 8043, 9688, 11732, 14066, 16963, 20260, 24310, 28953, 34586, 41047, 48857, 57802, 68528, 80862, 95534, 112388, 132391

Offset: 0

Emeric Deutsch, Sep 19 2016

nonn

The Heinz numbers of these partitions are given by A325130. - Gus Wiseman, Apr 02 2019

			a(4) = 3 because we have [1,1,1,1], [1,1,2], and [4]; the partitions [1,3], [2,2] do not qualify.
From _Gus Wiseman_, Apr 02 2019: (Start)
The a(2) = 2 through a(7) = 9 partitions:
  (2)   (3)    (4)     (5)      (6)       (7)
  (11)  (111)  (211)   (32)     (33)      (43)
               (1111)  (311)    (42)      (52)
                       (2111)   (222)     (511)
                       (11111)  (411)     (3211)
                                (3111)    (4111)
                                (21111)   (31111)
                                (111111)  (211111)
                                          (1111111)
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..12782 (terms 0..5000 from Alois P. Heinz)

Cf. A052335, A087153, A114639, A115584, A117144, A276427, A324572, A325130, A325131, A336269.

g := product(1/(1-x^i)-x^(i^2), i = 1 .. 100): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(i=j, 0, b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 19 2016

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[If[i == j, x, 1]*b[n - i*j, i - 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n][[1]], {n, 0, 60}] (* Jean-François Alcover, Nov 28 2016 after Alois P. Heinz's Maple code for A276427 *)
Table[Length[Select[IntegerPartitions[n],And@@Table[Count[#,i]!=i,{i,Union[#]}]&]],{n,0,30}] (* Gus Wiseman, Apr 02 2019 *)

a(n) = A276427(n,0).

G.f.: g(x) = Product_{i>=1} (1/(1-x^i) - x^{i^2}).

A242434 Number of compositions of n in which each part p has multiplicity p.

Original entry on oeis.org

1, 1, 0, 0, 1, 3, 0, 0, 0, 1, 4, 0, 0, 10, 60, 0, 1, 5, 0, 0, 15, 105, 0, 0, 0, 36, 286, 0, 0, 1281, 12768, 0, 0, 0, 56, 504, 1, 7, 2520, 27720, 28, 378, 1260, 0, 0, 7014, 84000, 0, 0, 4621, 83168, 360360, 210, 2346, 2522880, 37837800, 13860, 180180, 120, 1320

Offset: 0

Alois P. Heinz, May 14 2014

a(n) = 0 for n in {A001422}, a(n) > 0 for n in {A003995}.

			a(0) = 1: the empty composition.
a(1) = 1: [1].
a(4) = 1: [2,2].
a(5) = 3: [1,2,2], [2,1,2], [2,2,1].
a(9) = 1: [3,3,3].
a(10) = 4: [1,3,3,3], [3,1,3,3], [3,3,1,3], [3,3,3,1].
a(13) = 10: [2,2,3,3,3], [2,3,2,3,3], [2,3,3,2,3], [2,3,3,3,2], [3,2,2,3,3], [3,2,3,2,3], [3,2,3,3,2], [3,3,2,2,3], [3,3,2,3,2], [3,3,3,2,2].

Alois P. Heinz, Table of n, a(n) for n = 0..4500

Cf. A033461 (the same for partitions), A336269.

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
       b(n, i-1, p) +`if`(i^2>n, 0, b(n-i^2, i-1, p+i)/i!)))
    end:
a:= n-> b(n, isqrt(n), 0):
seq(a(n), n=0..100);

b[n_, i_, p_] := b[n, i, p] = If[n==0, p!, If[i<1, 0, b[n, i-1, p] + If[i^2 >n, 0, b[n-i^2, i-1, p+i]/i!]]]; a[n_] := b[n, Floor[Sqrt[n]], 0]; Table[ a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 08 2017, translated from Maple *)

A336273 Number of compositions of n containing at least one part p of multiplicity p.

Original entry on oeis.org

0, 1, 0, 2, 3, 5, 14, 28, 44, 101, 207, 399, 779, 1609, 3122, 6121, 11804, 23631, 46273, 91604, 178096, 352419, 691996, 1371306, 2702206, 5356324, 10604748, 21080216, 41869930, 83383786, 166114046, 331434088, 661685588, 1322042390, 2642367028, 5283397304

Offset: 0

Alois P. Heinz, Jul 15 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A011782, A336269.

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(`if`(i=j, 0, b(n-i*j, i-1, p+j)/j!), j=0..n/i)))
    end:
a:= n-> ceil(2^(n-1))-b(n$2, 0):
seq(a(n), n=0..40);

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0,
     Sum[If[i == j, 0, b[n - i*j, i - 1, p + j]/j!], {j, 0, n/i}]]];
a[n_] := Ceiling[2^(n - 1)] - b[n, n, 0];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 14 2022, after Alois P. Heinz *)

a(n) = A011782(n) - A336269(n).

cp's OEIS Frontend

A276429 Number of partitions of n containing no part i of multiplicity i.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A242434 Number of compositions of n in which each part p has multiplicity p.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A336273 Number of compositions of n containing at least one part p of multiplicity p.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula