A320818 Number of partitions of n with exactly five sorts of part 1 which are introduced in ascending order.

1, 15, 141, 1066, 7108, 43747, 255045, 1431320, 7814385, 41804990, 220266447, 1147232914, 5922585396, 30367092789, 154877631181, 786633449995, 3982378528296, 20109428513990, 101339359244739, 509871884291730, 2562078441467318, 12861324297841420

Offset: 5

Author

Alois P. Heinz, Oct 21 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 5..1433

Crossrefs

Column k=5 of A292746.

Cf. A320735, A320736.

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))(5):
seq(a(n), n=5..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 = 5}, b[n, n, k] - b[n, n, k - 1]];
a /@ Range[5, 35] (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)

Formula

a(n) = A320736(n) - A320735(n).