A117753 Triangle T(n, k) = f(n, 1 + (n mod 3))*f(k, 1 + (k mod 3)) mod n!, read by rows (see formula for f(n, k)).

0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 6, 6, 12, 6, 12, 0, 0, 0, 0, 0, 0, 24, 24, 48, 24, 144, 0, 576, 210, 210, 420, 210, 1260, 0, 0, 3780, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1728, 1728, 3456, 1728, 10368, 207360, 41472, 0, 0, 82944, 210, 210, 420, 210, 1260, 25200, 5040, 44100, 1209600, 362880, 44100

Offset

0,9

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = f(n, 1 + (n mod 3))*f(k, 1 + (k mod 3)) mod n!, where f(n, 1) = A049614(n), f(n, 2) = A034386(n), and f(n, 3) = n!.

Example

Triangle begins as:
     0;
     0,    0;
     0,    0,    0;
     1,    1,    2,    1;
     6,    6,   12,    6,    12;
     0,    0,    0,    0,     0,      0;
    24,   24,   48,   24,   144,      0,   576;
   210,  210,  420,  210,  1260,      0,     0,  3780;
     0,    0,    0,    0,     0,      0,     0,     0,   0;
  1728, 1728, 3456, 1728, 10368, 207360, 41472,     0,   0, 82944;

Mathematica

f[n_]:= If[PrimeQ[n], 1, n]; cf[n_]:= cf[n]= If[n==0, 1, f[n]*cf[n-1]]; (* A049614 *)
g[n_]:= If[PrimeQ[n], n, 1]; p[n_]:= p[n]= If[n==0, 1, g[n]*p[n-1]];  (* A034386 *)
f[n_, 1]= cf[n]; f[n_, 2]= p[n]; f[n_, 3]= n!;
Table[Mod[f[n, 1 + Mod[n, 3]]*f[m, 1 + Mod[m, 3]], n!], {n, 0, 10}, {m, 0, n}]//Flatten

Prog

(Magma)
A049614:= func< n | n le 1 select 1 else Factorial(n)/(&*[NthPrime(j): j in [1..#PrimesUpTo(n)]]) >;
A034386:= func< n | n eq 0 select 1 else LCM(PrimesInInterval(1, n)) >;
function f(n, k)
  if k eq 1 then return A049614(n);
  elif k eq 2 then return A034386(n);
  else return Factorial(n);
  end if;
end function;
A117753:= func< n, k | Floor( f( n, 1 + (n mod 3) )*f( k, 1 + (k mod 3)) ) mod Factorial(n) >;
[A117753(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 21 2023
(SageMath)
from sympy import primorial
def A049614(n): return factorial(n)/product(nth_prime(j) for j in range(1, 1+prime_pi(n)))
def A034386(n): return 1 if n == 0 else primorial(n, nth=False)
def f(n, m):
    if m==1: return A049614(n)
    elif m==2: return A034386(n)
    else: return factorial(n)
def A117753(n, k): return (f(n, 1+(n%3))*f(k, 1+(k%3)))%factorial(n)
flatten([[A117753(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 21 2023

Crossrefs

Cf. A034386, A049614, A117682.

Sequence in context: A111646 A363629 A342965 * A145883 A062820 A113336

Adjacent sequences: A117750 A117751 A117752 * A117754 A117755 A117756

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 14 2006

Extensions

Edited by G. C. Greubel, Jul 21 2023

Status

approved