A320816 Number of partitions of n with exactly three sorts of part 1 which are introduced in ascending order.

1, 6, 26, 97, 334, 1095, 3482, 10855, 33405, 101925, 309237, 934691, 2818110, 8482505, 25504000, 76625146, 230101961, 690759226, 2073184749, 6221368879, 18667736528, 56010470158, 168045932624, 504166843427, 1512558622966, 4537792056226, 13613608545770

Offset: 3

Author

Alois P. Heinz, Oct 21 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..2097

Crossrefs

Column k=3 of A292746.

Cf. A320733, A320734.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or i<2, add(
      Stirling2(n, j), j=0..k), add(b(n-i*j, i-1, k), j=0..n/i))
    end:
a:= n-> (k-> b(n$2, k)-b(n$2, k-1))(3):
seq(a(n), n=3..35);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i < 2, Sum[StirlingS2[n, j], {j, 0, k}], Sum[b[n - i*j, i - 1, k], {j, 0, n/i}]];
a[n_] := With[{k = 3}, b[n, n, k] - b[n, n, k - 1]];
a /@ Range[3, 35] (* Jean-François Alcover, Dec 16 2020, after Alois P. Heinz *)

Formula

a(n) = A320734(n) - A320733(n).