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

%I A258460 #12 Oct 15 2018 10:34:14
%S A258460 1,16,157,1223,8331,52078,307122,1738441,9552809,51357781,271624053,
%T A258460 1418856775,7341440755,37708531955,192586153199,979219591861,
%U A258460 4961598056587,25071026497266,126410385360189,636282269208285,3198360708483673,16059685003763157
%N A258460 Number of partitions of n into parts of exactly 5 sorts which are introduced in ascending order.
%H A258460 Alois P. Heinz, <a href="/A258460/b258460.txt">Table of n, a(n) for n = 5..1000</a>
%F A258460 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
%p A258460 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A258460       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
%p A258460     end:
%p A258460 T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
%p A258460 a:= n-> T(n,5):
%p A258460 seq(a(n), n=5..30);
%t A258460 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 A258460 T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i/(i!*(k - i)!), {i, 0, k}];
%t A258460 a[n_] := T[n, 5];
%t A258460 Table[a[n], {n, 5, 30}] (* _Jean-François Alcover_, May 22 2018, translated from Maple *)
%Y A258460 Column k=5 of A256130.
%Y A258460 Cf. A320547.
%K A258460 nonn
%O A258460 5,2
%A A258460 _Alois P. Heinz_, May 30 2015