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A163402 A Minkowski-type generalization of the factorial function: F(n,k) with k = 2.

%I A163402 #11 Feb 27 2020 11:53:49
%S A163402 1,1,1,3,9,135,1215,2835,127575,229635,3444525,1705039875,
%T A163402 107417512125,13299311025,4189282972875,62839244593125,
%U A163402 188517733779375,336504154796184375,9085612179496978125,2740105260483215625
%N A163402 A Minkowski-type generalization of the factorial function: F(n,k) with k = 2.
%C A163402 F(n,0) = n! (A000142)
%C A163402 F(n,1) = Minkowski(n)/n! (A163176)
%C A163402 F(n,2) = a(n)
%F A163402 P(n,k) = {p prime | k+1 <= p <= n }
%F A163402 L(n,p,r) = Sum_{i>=0} floor((n-r)/((p-r)*p^i))
%F A163402 A(n,k) = Prod_{p in P(n,k)} p^(Sum_{m=0..k} (-1)^m*L(n,p,m))
%F A163402 F(n,k) = A(n,k)^((-1)^k).
%e A163402 For n >= 0
%e A163402 F(n,0) 1, 1, 2, 6, 24, 120, 720, 5040, 40320, ...
%e A163402 F(n,1) 1, 1, 1, 4, 2, 48, 16, 576, 144, 3840, ...
%e A163402 F(n,2) 1, 1, 1, 3, 9, 135, 1215, 2835, 127575, ...
%e A163402 F(n,3) 1, 1, 1, 1, 1, 1, 1, 5, 1, 25, 5, 35, ...
%e A163402 F(n,4) 1, 1, 1, 1, 1, 5, 25, 175, 4375, 4375, ...
%e A163402 F(n,5) 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 49, ...
%p A163402 F := proc(n,k) local L,p,i;
%p A163402 L := proc(n,u,r) local q,s,m; m:=n-r;
%p A163402 q:=u-r; s:=0; do if q>m then break fi;
%p A163402 s:=s+iquo(m,q); q:=q*u od; s end;
%p A163402 mul(p^add((-1)^i*L(n,p,i),i=0..k),
%p A163402 p = select(isprime,[$(k+1)..n]))^((-1)^k) end:
%p A163402 a(n) := n -> F(n,2);
%t A163402 F[n_, k_] := Module[{L, p, i}, L[n0_, u_, r_] := Module[{q, s, m}, m = n0-r; q = u-r; s = 0; While[True, If[q > m, Break[]]; s = s + Quotient[m, q]; q = q*u]; s]; Product[p^Sum[(-1)^i*L[n, p, i], {i, 0, k}], {p, Select[Range[k+1, n], PrimeQ]}]^((-1)^k)]; a[n_] := F[n, 2]; Table[a[n], {n, 0, 19}] (* _Jean-François Alcover_, Jan 15 2014, translated from Maple *)
%o A163402 (Sage)
%o A163402 def A163402(n):
%o A163402     def L(n, u, r):
%o A163402         m = n - r; q = u - r
%o A163402         s = 0
%o A163402         while(q <= m):
%o A163402             s += m//q
%o A163402             q *= u
%o A163402         return s
%o A163402     P = filter(is_prime, [3..n])
%o A163402     return mul(p^add((-1)^i*L(n, p, i) for i in (0..2)) for p in P)
%o A163402 print([A163402(n) for n in range(20)]) # _Peter Luschny_, Mar 13 2016
%Y A163402 Cf. A000142, A053657, A163176.
%K A163402 nonn
%O A163402 0,4
%A A163402 _Peter Luschny_, Jul 26 2009