A320545 Number of partitions of n into parts of exactly three sorts which are introduced in ascending order such that sorts of adjacent parts are different.

1, 4, 12, 31, 73, 165, 357, 760, 1582, 3270, 6678, 13589, 27482, 55468, 111588, 224259, 449908, 902106, 1807173, 3619162, 7244557, 14499238, 29011551, 58044194, 116115782, 232275383, 464607730, 929306306, 1858730674, 3717648658, 7435541392, 14871467784

Offset: 3

Author

Alois P. Heinz, Oct 15 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..1000

Crossrefs

Column k=3 of A262495.

Cf. A258458, A262445.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
a:= n-> (k-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k))(3):
seq(a(n), n=3..40);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, k^(n - 1), b[n, i - 1, k] + If[i > n, 0, k b[n - i, i, k]]];
A[n_, k_] := If[n == 0, 1, If[k < 2, k, k b[n, n, k - 1]]];
a[n_] := With[{k = 3}, Sum[A[n, k - i] (-1)^i/(i! (k - i)!), {i, 0, k}]];
a /@ Range[3, 40] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)

Formula

a(n) ~ 2^(n-2) / QPochhammer[1/2]. - Vaclav Kotesovec, Oct 25 2018