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

3,2

Links

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

Formula

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

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 *)

Crossrefs

Column k=3 of A292746.

Cf. A320733, A320734.

Sequence in context: A143132 A055589 A318947 * A239179 A307309 A186314

Adjacent sequences: A320813 A320814 A320815 * A320817 A320818 A320819

Keyword

nonn

Author

Alois P. Heinz, Oct 21 2018

Status

approved