A365617 Iterated Pochhammer symbol.
1, 1, 2, 24, 421200, 13257209623458438290962108800
Offset: 0
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..6
- Wikipedia, Falling and rising factorials
Programs
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Maple
a:= proc(n) option remember; `if`(n<0, 0, (z-> mul(z+j, j=0..n-1))(a(n-1))) end: seq(a(n), n=0..5); # Alois P. Heinz, Sep 12 2023 -
Mathematica
FoldList[Pochhammer, 1, Range[5]] (* Amiram Eldar, Sep 12 2023 *)
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PARI
P(x, y) = my(P=1); for (i=0, y-1, P *= x+i); P; a(n) = my(x=1); n--; for (i=1, n, x = P(x, i+1)); x; \\ Michel Marcus, Sep 13 2023
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Python
from gmpy2 import * from functools import reduce gamma = lambda n: fac(n - 1) Pochhammer = lambda z,n: gamma(n + z) // gamma(z) list_Pochhammer = lambda lst: int(reduce((lambda x, y: Pochhammer(x, y)), lst)) if len(lst) > 0 else 1 print([list_Pochhammer(range(1, n + 1)) for n in range(0, 6)])
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Python
from functools import reduce from sympy import rf def A365617(n): return reduce(rf,range(1,n+1),1) # Chai Wah Wu, Sep 15 2023
Formula
a(n) = Pochhammer(...(Pochhammer(Pochhammer(1, 2), 3), ...), n).
a(n) = gamma(n + a(n-1)) / gamma(a(n-1)).
a(n) = Product_{j=0..n-1} (j + a(n-1)), a(0) = 1. - Alois P. Heinz, Sep 12 2023