A258461 Number of partitions of n into parts of exactly 6 sorts which are introduced in ascending order.

1, 22, 289, 2957, 26073, 208516, 1558219, 11087756, 76079368, 507834013, 3318628444, 21330627775, 135325210699, 849659799754, 5290544981423, 32722489513367, 201296535378562, 1232850239039750, 7523511821431264, 45777353199866275, 277862479920868778

Offset: 6

Author

Alois P. Heinz, May 30 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 6..1000

Crossrefs

Column k=6 of A256130.

Cf. A320548.

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,6):
seq(a(n), n=6..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}];
Table[T[n, 6], {n, 6, 30}] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * 6^n, where c = 1/(6!*Product_{n>=1} (1-1/6^n)) = 1/(6!*QPochhammer[1/6, 1/6]) = 0.001723855087202395653855120059043... . - Vaclav Kotesovec, Jun 01 2015