A258460 Number of partitions of n into parts of exactly 5 sorts which are introduced in ascending order.

1, 16, 157, 1223, 8331, 52078, 307122, 1738441, 9552809, 51357781, 271624053, 1418856775, 7341440755, 37708531955, 192586153199, 979219591861, 4961598056587, 25071026497266, 126410385360189, 636282269208285, 3198360708483673, 16059685003763157

Offset: 5

Author

Alois P. Heinz, May 30 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 5..1000

Crossrefs

Column k=5 of A256130.

Cf. A320547.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> T(n,5):
seq(a(n), n=5..30);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k*b[n - i, i, k]]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i/(i!*(k - i)!), {i, 0, k}];
a[n_] := T[n, 5];
Table[a[n], {n, 5, 30}] (* Jean-François Alcover, May 22 2018, translated from Maple *)

Formula

a(n) ~ c * 5^n, where c = 1/(5!*Product_{n>=1} (1-1/5^n)) = 1/(5!*QPochhammer[1/5, 1/5]) = 1/(5!*A100222) = 0.0109601129644612101609007882... . - Vaclav Kotesovec, Jun 01 2015