A114806 Nonuple factorial, 9-factorial, n!9, n!!!!!!!!!.

1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 22, 36, 52, 70, 90, 112, 136, 162, 190, 440, 756, 1144, 1610, 2160, 2800, 3536, 4374, 5320, 12760, 22680, 35464, 51520, 71280, 95200, 123760, 157464, 196840, 484880, 884520, 1418560, 2112320, 2993760, 4093600

Offset: 0

Author

Jonathan Vos Post, Feb 19 2006

Keywords

• easy
• nonn

Examples

			a(10) = 10 * a(10-9) = 10 * a(1) = 10 * 1 = 10.
a(20) = 20 * a(20-9) = 20 * a(11) = 20 * (11*a(11-9)) = 20 * 11 * a(2) = 20 * 11 * 2 = 440.
a(30) = 30 * a(30-9) = 30 * a(21) = 30 * (21*a(21-9)) = 30 * 21 * a(12) = 30 * 21 * (12*a(12-9)) = 30 * 21 * 12 * 3 = 22680.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..2955
• Eric Weisstein's World of Mathematics, Multifactorial.

Crossrefs

Cf. A000142, A006882, A007661, A007662, A085157, A085158, A288096.

Programs

GAP

a:= function(n)
    if n<1 then return 1;
    else return n*a(n-9);
    fi;
  end;
List([0..50], n-> a(n) ); # G. C. Greubel, Aug 21 2019

Magma

b:=func< n | n le 9 select n else n*Self(n-9) >;
[1] cat [b(n): n in [1..50]]; // G. C. Greubel, Aug 21 2019

Maple

f:= proc(n) option remember;
n*procname(n-9)
end proc:
f(0):= 1: for n from 1 to 8 do f(n):= n od:
map(f, [$0..100]); # Robert Israel, Jun 21 2019

Mathematica

NFactorialM[n_, m_] := Block[{k = n, p = Max[1, n]}, While[k > m, k -= m; p *= k]; p]; Array[ NFactorialM[#, 9] &, 44, 0] (* Robert G. Wilson v, May 10 2011 *)
a[n_]:= a[n]= If[n<1, 1, n*a[n-9]]; Table[a[n], {n,0,50}] (* G. C. Greubel, Aug 21 2019 *)
Table[Times@@Range[n,1,-9],{n,0,50}] (* Harvey P. Dale, Nov 13 2021 *)

PARI

a(n)=if(n<1, 1, n*a(n-9));
vector(50, n, n--; a(n) ) \\ G. C. Greubel, Aug 21 2019

Sage

def a(n):
    if (n<1): return 1
    else: return n*a(n-9)
[a(n) for n in (0..50)] # G. C. Greubel, Aug 21 2019

Formula

D-finite with recurrence: a(0) = 1, a(n) = n for 1 <= n <= 9, a(n) = n*a(n-9) for n >= 10.

From Robert Israel, Jun 21 2019: (Start)

a(9*m) = 9^m*m!.

a(9*m+k) = 9^m*(9*m+k)*Gamma(m+k/9)/Gamma(k/9) for 1 <= k <= 8. (End)

Sum_{n>=0} 1/a(n) = A288096. - Amiram Eldar, Nov 10 2020