A117683 Triangle T(n,k) = A049614(n)/(A049614(k)*A049614(n-k)), read by rows.

1, 1, 1, 1, 1, 1, 4, 4, 4, 1, 1, 4, 4, 1, 1, 6, 6, 24, 6, 6, 1, 1, 6, 6, 6, 6, 1, 1, 8, 8, 48, 12, 48, 8, 8, 1, 9, 72, 72, 108, 108, 72, 72, 9, 1, 10, 90, 720, 180, 1080, 180, 720, 90, 10, 1, 1, 10, 90, 180, 180, 180, 180, 90, 10, 1, 1, 12, 12, 120, 270, 2160, 360, 2160, 270, 120, 12, 12, 1

Offset

1,7

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Formula

T(n,k) = A049614(n)/(A049614(k)*A049614(n-k)), for 1 <= k <= n, n >= 1.

Sum_{k=1..n} T(n, k) = A117684(n).

Example

Triangle begins as:
  1;
  1,  1;
  1,  1,  1;
  4,  4,  4,   1;
  1,  4,  4,   1,   1;
  6,  6, 24,   6,   6,  1;
  1,  6,  6,   6,   6,  1,  1;
  8,  8, 48,  12,  48,  8,  8,  1;
  9, 72, 72, 108, 108, 72, 72,  9,  1;

Mathematica

f[n_]:= If[PrimeQ[n], 1, n];
cf[n_]:= cf[n]= If[n==0, 1, f[n]*cf[n-1]]; (* A049614 *)
T[n_, k_]:= T[n, k]= cf[n]/(cf[k]*cf[n-k]);
Table[T[n, k], {n, 12}, {k, n}]//Flatten

Prog

(PARI) primorial(n)=prod(i=1, primepi(n), prime(i))
T(n, m)=binomial(n, m)*primorial(m)*primorial(n-m)/primorial(n) \\ Charles R Greathouse IV, Jan 16 2012
(Magma)
A049614:= func< n | n le 1 select 1 else Factorial(n)/(&*[NthPrime(j): j in [1..#PrimesUpTo(n)]]) >;
A117683:= func< n, k | A049614(n)/(A049614(k)*A049614(n-k)) >;
[A117683(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 21 2023
(SageMath)
def A049614(n): return factorial(n)/product(nth_prime(j) for j in range(1, 1+prime_pi(n)))
def A117683(n, k): return A049614(n)/(A049614(k)*A049614(n-k))
flatten([[A117683(n, k) for k in range(1, n+1)] for n in range(1, 13)]) # G. C. Greubel, Jul 21 2023

Crossrefs

Cf. A049614, A117684.

Sequence in context: A133889 A243756 A172985 * A172088 A159891 A201941

Adjacent sequences: A117680 A117681 A117682 * A117684 A117685 A117686

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 12 2006

Extensions

Edited by the Associate Editors of the OEIS, Aug 18 2009

Edited by G. C. Greubel, Jul 21 2023

Status

approved