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

0,7

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;
     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;

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

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;
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
(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

Crossrefs

Cf. A034386, A049614, A117682, A117753.

Sequence in context: A051886 A244080 A141652 * A248577 A015999 A225244

Adjacent sequences: A117751 A117752 A117753 * A117755 A117756 A117757

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 14 2006

Extensions

Edited by G. C. Greubel, Jul 21 2023

Status

approved