A258464 Number of partitions of n into parts of exactly 9 sorts which are introduced in ascending order.

1, 46, 1202, 23523, 384227, 5542879, 73055550, 899381476, 10501235760, 117575627562, 1272685923724, 13401470756233, 137945728220761, 1393299928219604, 13851195993228228, 135865787060383171, 1317624915100561406, 12654868264707446322, 120534359759023523561

Offset: 9

Author

Alois P. Heinz, May 30 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 9..1000

Crossrefs

Column k=9 of A256130.

Cf. A320551.

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,9):
seq(a(n), n=9..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, 9], {n, 9, 30}] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * 9^n, where c = 1/(9!*Product_{n>=1} (1-1/9^n)) = 1/(9!*QPochhammer[1/9, 1/9]) = 0.0000031438016899923866898607402658778352... . - Vaclav Kotesovec, Jun 01 2015