A365617 - OEIS

%I #50 Sep 15 2023 16:24:21

%S 1,1,2,24,421200,13257209623458438290962108800

%N Iterated Pochhammer symbol.

%H Alois P. Heinz, <a href="/A365617/b365617.txt">Table of n, a(n) for n = 0..6</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Falling_and_rising_factorials">Falling and rising factorials</a>

%F a(n) = Pochhammer(...(Pochhammer(Pochhammer(1, 2), 3), ...), n).

%F a(n) = gamma(n + a(n-1)) / gamma(a(n-1)).

%F a(n) = Product_{j=0..n-1} (j + a(n-1)), a(0) = 1. - _Alois P. Heinz_, Sep 12 2023

%p a:= proc(n) option remember; `if`(n<0, 0,

%p       (z-> mul(z+j, j=0..n-1))(a(n-1)))

%p     end:

%p seq(a(n), n=0..5);  # _Alois P. Heinz_, Sep 12 2023

%t FoldList[Pochhammer, 1, Range[5]] (* _Amiram Eldar_, Sep 12 2023 *)

%o (Python)

%o from gmpy2 import *

%o from functools import reduce

%o gamma = lambda n: fac(n - 1)

%o Pochhammer = lambda z,n: gamma(n + z) // gamma(z)

%o list_Pochhammer = lambda lst: int(reduce((lambda x, y: Pochhammer(x, y)), lst)) if len(lst) > 0 else 1

%o print([list_Pochhammer(range(1, n + 1)) for n in range(0, 6)])

%o (Python)

%o from functools import reduce

%o from sympy import rf

%o def A365617(n): return reduce(rf,range(1,n+1),1) # _Chai Wah Wu_, Sep 15 2023

%o (PARI) P(x, y) = my(P=1); for (i=0, y-1, P *= x+i); P;	

%o a(n) = my(x=1); n--; for (i=1, n, x = P(x, i+1)); x; \\ _Michel Marcus_, Sep 13 2023

%Y Cf. A000142, A000407, A038155, A055462, A112332, A266083.

%K nonn

%O 0,3

%A _Darío Clavijo_, Sep 12 2023