A117754 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, 1, 1, 0, 2, 2, 3, 2, 7, 7, 8, 7, 12, 1, 1, 2, 1, 6, 0, 25, 25, 26, 25, 30, 144, 48, 211, 211, 212, 211, 216, 330, 234, 420, 1, 1, 2, 1, 6, 120, 24, 210, 0, 1729, 1729, 1730, 1729, 1734, 1848, 1752, 1938, 42048, 3456, 211, 211, 212, 211, 216, 330, 234, 420, 40530, 1938, 420

Offset: 0

Author

Roger L. Bagula, Apr 14 2006

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
     0;
     0,    0;
     1,    1,    0;
     2,    2,    3,    2;
     7,    7,    8,    7,   12;
     1,    1,    2,    1,    6,    0;
    25,   25,   26,   25,   30,  144,   48;
   211,  211,  212,  211,  216,  330,  234,  420;
     1,    1,    2,    1,    6,  120,   24,  210,     0;
  1729, 1729, 1730, 1729, 1734, 1848, 1752, 1938, 42048, 3456;
   211,  211,  212,  211,  216,  330,  234,  420, 40530, 1938, 420;
		

Links

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

Crossrefs

Cf. A034386, A049614, A117682, A117753.

Programs

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;
A117754:= func< n,k | Floor(f(n, 1+(n mod 3))+f( k, 1+(k mod 3))) mod
 Factorial(n) >;
[A117754(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 21 2023

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!;
T[n_, k_]:= Mod[f[n, 1 + Mod[n, 3]] + f[k, 1 + Mod[k, 3]], n!];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten

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 A117754(n, k): return (f(n, 1+(n%3))+f(k, 1+(k%3)))%factorial(n)
flatten([[A117754(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 21 2023

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!.

Extensions

Edited by G. C. Greubel, Jul 21 2023