cp's OEIS Frontend

A365617 Iterated Pochhammer symbol.

%I A365617 #53 Apr 27 2025 00:45:56
%S A365617 1,1,2,24,421200,13257209623458438290962108800
%N A365617 Iterated Pochhammer symbol.
%H A365617 Alois P. Heinz, <a href="/A365617/b365617.txt">Table of n, a(n) for n = 0..6</a>
%H A365617 Wikipedia, <a href="https://en.wikipedia.org/wiki/Falling_and_rising_factorials">Falling and rising factorials</a>
%F A365617 a(n) = Pochhammer(...(Pochhammer(Pochhammer(1, 2), 3), ...), n).
%F A365617 a(n) = gamma(n + a(n-1)) / gamma(a(n-1)).
%F A365617 a(n) = Product_{j=0..n-1} (j + a(n-1)), a(0) = 1. - _Alois P. Heinz_, Sep 12 2023
%p A365617 a:= proc(n) option remember; `if`(n<0, 0,
%p A365617       (z-> mul(z+j, j=0..n-1))(a(n-1)))
%p A365617     end:
%p A365617 seq(a(n), n=0..5);  # _Alois P. Heinz_, Sep 12 2023
%t A365617 FoldList[Pochhammer, 1, Range[5]] (* _Amiram Eldar_, Sep 12 2023 *)
%o A365617 (Python)
%o A365617 from gmpy2 import *
%o A365617 from functools import reduce
%o A365617 gamma = lambda n: fac(n - 1)
%o A365617 Pochhammer = lambda z,n: gamma(n + z) // gamma(z)
%o A365617 list_Pochhammer = lambda lst: int(reduce((lambda x, y: Pochhammer(x, y)), lst)) if len(lst) > 0 else 1
%o A365617 print([list_Pochhammer(range(1, n + 1)) for n in range(0, 6)])
%o A365617 (Python)
%o A365617 from functools import reduce
%o A365617 from sympy import rf
%o A365617 def A365617(n): return reduce(rf,range(1,n+1),1) # _Chai Wah Wu_, Sep 15 2023
%o A365617 (PARI) P(x, y) = my(P=1); for (i=0, y-1, P *= x+i); P;
%o A365617 a(n) = my(x=1); n--; for (i=1, n, x = P(x, i+1)); x; \\ _Michel Marcus_, Sep 13 2023
%Y A365617 Cf. A000142, A000407, A038155, A055462, A112332, A266083.
%K A365617 nonn
%O A365617 0,3
%A A365617 _Darío Clavijo_, Sep 12 2023