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A141617 Triangle read by rows: T(n, k) = binomial(n,k)*prime(k)*prime(n-k), for 1 <= k <= n-1, n >= 1, with T(0, 0) = 1, T(n, 0) = T(n, n) = prime(n).

%I A141617 #15 Oct 26 2024 10:45:08
%S A141617 1,2,2,3,8,3,5,18,18,5,7,40,54,40,7,11,70,150,150,70,11,13,132,315,
%T A141617 500,315,132,13,17,182,693,1225,1225,693,182,17,19,272,1092,3080,3430,
%U A141617 3080,1092,272,19,23,342,1836,5460,9702,9702,5460,1836,342,23
%N A141617 Triangle read by rows: T(n, k) = binomial(n,k)*prime(k)*prime(n-k), for 1 <= k <= n-1, n >= 1, with T(0, 0) = 1, T(n, 0) = T(n, n) = prime(n).
%C A141617 For the purpose of this sequence define prime(0)=1.
%H A141617 G. C. Greubel, <a href="/A141617/b141617.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A141617 Symmetry: T(n, k) = T(n, n-k).
%e A141617 Triangle begins as:
%e A141617    1;
%e A141617    2,   2;
%e A141617    3,   8,    3;
%e A141617    5,  18,   18,     5;
%e A141617    7,  40,   54,    40,     7;
%e A141617   11,  70,  150,   150,    70,    11;
%e A141617   13, 132,  315,   500,   315,   132,    13;
%e A141617   17, 182,  693,  1225,  1225,   693,   182,    17;
%e A141617   19, 272, 1092,  3080,  3430,  3080,  1092,   272,   19;
%e A141617   23, 342, 1836,  5460,  9702,  9702,  5460,  1836,  342,  23;
%e A141617   29, 460, 2565, 10200, 19110, 30492, 19110, 10200, 2565, 460, 29;
%e A141617   ...
%p A141617 p:= n-> `if`(n=0, 1, ithprime(n)):
%p A141617 T:= (n, k)-> binomial(n, k)*p(k)*p(n-k):
%p A141617 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Apr 26 2023
%t A141617 A141617[n_, k_]:= If[n==0, 1, If[k==0 || k==n, Prime[n], Binomial[n, k]*Prime[k]*Prime[n-k]]];
%t A141617 Table[A414617[n,k], {n,0,12}, {k,0,n}]//Flatten
%o A141617 (Magma)
%o A141617 function A141617(n,k)
%o A141617   if n eq 0 then return 1;
%o A141617   else return Binomial(n,k)*NthPrime(k)*NthPrime(n-k);
%o A141617   end if;
%o A141617 end function;
%o A141617 [A141617(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 26 2024
%o A141617 (SageMath)
%o A141617 def A141617(n,k):
%o A141617     if n==0: return 1
%o A141617     elif k==0 or k==n: return nth_prime(n)
%o A141617     else: return binomial(n,k)*nth_prime(k)*nth_prime(n-k)
%o A141617 flatten([[A141617(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Oct 26 2024
%Y A141617 Cf. A008578, A098350.
%K A141617 nonn,easy,tabl
%O A141617 0,2
%A A141617 _Roger L. Bagula_ and _Gary W. Adamson_, Aug 23 2008