login
A117683
Triangle T(n,k) = A049614(n)/(A049614(k)*A049614(n-k)), read by rows.
6
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
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
Sequence in context: A133889 A243756 A172985 * A172088 A159891 A201941
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