A244239 Number of partitions of n into 3 parts such that every i-th smallest part (counted with multiplicity) is different from i.

1, 3, 4, 6, 7, 9, 11, 13, 15, 18, 20, 23, 26, 29, 32, 36, 39, 43, 47, 51, 55, 60, 64, 69, 74, 79, 84, 90, 95, 101, 107, 113, 119, 126, 132, 139, 146, 153, 160, 168, 175, 183, 191, 199, 207, 216, 224, 233, 242, 251, 260, 270, 279, 289, 299, 309, 319, 330, 340

Offset: 9

Author

Alois P. Heinz, Jun 23 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 9..1000
• N. Guru Sharan and Armin Straub, Partitions with Durfee triangles of fixed size, arXiv:2507.19047 [math.CO], 2025. See p. 5.
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Crossrefs

Column k=3 of A238406.

Programs

Maple

a:= proc(n) option remember; `if`(n<14, [1, 3, 4, 6, 7][n-8],
      ((-4*n+56)*a(n-5) +(3*n-16)*a(n-4) +(7*n-66)*a(n-3)
      +(4*n-44)*a(n-2) +(28-3*n)*a(n-1)) / (7*n-78))
    end:
seq(a(n), n=9..80);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, Function[p, Expand[x*(p-Coefficient[p, x, i-1]*x^(i-1))]][b[n-i, i]]]]];
a[n_] := Coefficient[b[n, n], x, 3];
Table[a[n], {n, 9, 80}] (* Jean-François Alcover, May 01 2018, after Alois P. Heinz *)

Formula

From Chai Wah Wu, Apr 18 2024: (Start)

a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n > 15.

G.f.: x^9*(-x^6 + 2*x^4 + x^3 - 2*x - 1)/((x - 1)^3*(x + 1)*(x^2 + x + 1)). (End)