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

%I A258461 #12 Dec 07 2020 14:00:54
%S A258461 1,22,289,2957,26073,208516,1558219,11087756,76079368,507834013,
%T A258461 3318628444,21330627775,135325210699,849659799754,5290544981423,
%U A258461 32722489513367,201296535378562,1232850239039750,7523511821431264,45777353199866275,277862479920868778
%N A258461 Number of partitions of n into parts of exactly 6 sorts which are introduced in ascending order.
%H A258461 Alois P. Heinz, <a href="/A258461/b258461.txt">Table of n, a(n) for n = 6..1000</a>
%F A258461 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
%p A258461 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A258461       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
%p A258461     end:
%p A258461 T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
%p A258461 a:= n-> T(n,6):
%p A258461 seq(a(n), n=6..30);
%t A258461 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 A258461 T[n_, k_] := Sum[b[n, n, k - i](-1)^i/(i!(k - i)!), {i, 0, k}];
%t A258461 Table[T[n, 6], {n, 6, 30}] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y A258461 Column k=6 of A256130.
%Y A258461 Cf. A320548.
%K A258461 nonn
%O A258461 6,2
%A A258461 _Alois P. Heinz_, May 30 2015