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A347611 a(n) is the n-th n-factorial number: a(n) = n!_n.

%I A347611 #27 Jun 09 2025 06:35:13
%S A347611 1,1,3,52,8925,22661496,1131162092095,1375009641495014400,
%T A347611 48378633136349277767794425,57001313848230245122464621625840000,
%U A347611 2552524038347870310755413660544832496799359491,4859161865915056755501262525796512204608930674134393036800
%N A347611 a(n) is the n-th n-factorial number: a(n) = n!_n.
%H A347611 Alois P. Heinz, <a href="/A347611/b347611.txt">Table of n, a(n) for n = 0..36</a>
%H A347611 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A347611 a(n) = Product_{j=1..n} (n^j-1)/(n-1) for n > 1, a(0) = a(1) = 1.
%F A347611 a(n) = A069777(n,n).
%F A347611 a(n) ~ exp(1) * n^(n*(n-1)/2). - _Vaclav Kotesovec_, Jun 09 2025
%p A347611 b:= proc(n, k) option remember; `if`(n<2, 1,
%p A347611       b(n-1, k)*(k^n-1)/(k-1))
%p A347611     end:
%p A347611 a:= n-> b(n$2):
%p A347611 seq(a(n), n=0..12);
%t A347611 Array[QFactorial[#, #] &, 12, 0] (* _Michael De Vlieger_, Sep 09 2021 *)
%o A347611 (PARI) a(n) = if (n<=1, 1, prod(k=1, n, (n^k-1)/(n-1))); \\ _Michel Marcus_, Sep 09 2021
%o A347611 (Python)
%o A347611 from math import prod
%o A347611 def a(n):
%o A347611     return 1 if n <= 1 else prod((n**k - 1)//(n - 1) for k in range(1, n+1))
%o A347611 print([a(n) for n in range(12)]) # _Michael S. Branicky_, Sep 09 2021
%Y A347611 Main diagonal of A069777.
%Y A347611 Cf. A366355.
%K A347611 nonn
%O A347611 0,3
%A A347611 _Alois P. Heinz_, Sep 08 2021