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

3,2

Links

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

Formula

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

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 *)

Crossrefs

Column k=3 of A262495.

Cf. A258458, A262445.

Sequence in context: A027658 A001982 A129707 * A232580 A133546 A190376

Adjacent sequences: A320542 A320543 A320544 * A320546 A320547 A320548

Keyword

nonn

Author

Alois P. Heinz, Oct 15 2018

Status

approved