A320817 Number of partitions of n with exactly four sorts of part 1 which are introduced in ascending order.

1, 10, 66, 361, 1778, 8207, 36310, 156095, 657785, 2733065, 11241497, 45900679, 186420826, 754165809, 3042167236, 12245294090, 49211278321, 197535872510, 792216674789, 3175088068035, 12719020008668, 50932090504830, 203896407951944, 816089798651203

Offset: 4

Author

Alois P. Heinz, Oct 21 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 4..1663

Crossrefs

Column k=4 of A292746.

Cf. A320734, A320735.

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))(4):
seq(a(n), n=4..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 = 4}, b[n, n, k] - b[n, n, k-1]];
a /@ Range[4, 35] (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)

Formula

a(n) = A320735(n) - A320734(n).