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

%I A117683 #14 Jul 22 2023 12:12:07
%S A117683 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,
%T A117683 48,8,8,1,9,72,72,108,108,72,72,9,1,10,90,720,180,1080,180,720,90,10,
%U A117683 1,1,10,90,180,180,180,180,90,10,1,1,12,12,120,270,2160,360,2160,270,120,12,12,1
%N A117683 Triangle T(n,k) = A049614(n)/(A049614(k)*A049614(n-k)), read by rows.
%H A117683 G. C. Greubel, <a href="/A117683/b117683.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A117683 T(n,k) = A049614(n)/(A049614(k)*A049614(n-k)), for 1 <= k <= n, n >= 1.
%F A117683 Sum_{k=1..n} T(n, k) = A117684(n).
%e A117683 Triangle begins as:
%e A117683   1;
%e A117683   1,  1;
%e A117683   1,  1,  1;
%e A117683   4,  4,  4,   1;
%e A117683   1,  4,  4,   1,   1;
%e A117683   6,  6, 24,   6,   6,  1;
%e A117683   1,  6,  6,   6,   6,  1,  1;
%e A117683   8,  8, 48,  12,  48,  8,  8,  1;
%e A117683   9, 72, 72, 108, 108, 72, 72,  9,  1;
%t A117683 f[n_]:= If[PrimeQ[n], 1, n];
%t A117683 cf[n_]:= cf[n]= If[n==0, 1, f[n]*cf[n-1]]; (* A049614 *)
%t A117683 T[n_, k_]:= T[n, k]= cf[n]/(cf[k]*cf[n-k]);
%t A117683 Table[T[n, k], {n,12}, {k,n}]//Flatten
%o A117683 (PARI) primorial(n)=prod(i=1,primepi(n),prime(i))
%o A117683 T(n,m)=binomial(n,m)*primorial(m)*primorial(n-m)/primorial(n) \\ _Charles R Greathouse IV_, Jan 16 2012
%o A117683 (Magma)
%o A117683 A049614:= func< n | n le 1 select 1 else Factorial(n)/(&*[NthPrime(j): j in [1..#PrimesUpTo(n)]]) >;
%o A117683 A117683:= func< n,k | A049614(n)/(A049614(k)*A049614(n-k)) >;
%o A117683 [A117683(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jul 21 2023
%o A117683 (SageMath)
%o A117683 def A049614(n): return factorial(n)/product(nth_prime(j) for j in range(1, 1+prime_pi(n)))
%o A117683 def A117683(n,k): return A049614(n)/(A049614(k)*A049614(n-k))
%o A117683 flatten([[A117683(n,k) for k in range(1,n+1)] for n in range(1,13)]) # _G. C. Greubel_, Jul 21 2023
%Y A117683 Cf. A049614, A117684.
%K A117683 nonn,tabl
%O A117683 1,7
%A A117683 _Roger L. Bagula_, Apr 12 2006
%E A117683 Edited by the Associate Editors of the OEIS, Aug 18 2009
%E A117683 Edited by _G. C. Greubel_, Jul 21 2023