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
Keywords
Examples
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)
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..12782 (terms 0..5000 from Alois P. Heinz)
Crossrefs
Programs
-
Maple
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 -
Mathematica
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 *)
Formula
a(n) = A276427(n,0).
G.f.: g(x) = Product_{i>=1} (1/(1-x^i) - x^{i^2}).
Comments