%I #12 Dec 07 2020 14:00:18
%S 1,46,1202,23523,384227,5542879,73055550,899381476,10501235760,
%T 117575627562,1272685923724,13401470756233,137945728220761,
%U 1393299928219604,13851195993228228,135865787060383171,1317624915100561406,12654868264707446322,120534359759023523561
%N Number of partitions of n into parts of exactly 9 sorts which are introduced in ascending order.
%H Alois P. Heinz, <a href="/A258464/b258464.txt">Table of n, a(n) for n = 9..1000</a>
%F 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
%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
%p end:
%p T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
%p a:= n-> T(n,9):
%p seq(a(n), n=9..30);
%t 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 T[n_, k_] := Sum[b[n, n, k - i] (-1)^i/(i! (k - i)!), {i, 0, k}];
%t Table[T[n, 9], {n, 9, 30}] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y Column k=9 of A256130.
%Y Cf. A320551.
%K nonn
%O 9,2
%A _Alois P. Heinz_, May 30 2015