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).

1, 2, 2, 3, 8, 3, 5, 18, 18, 5, 7, 40, 54, 40, 7, 11, 70, 150, 150, 70, 11, 13, 132, 315, 500, 315, 132, 13, 17, 182, 693, 1225, 1225, 693, 182, 17, 19, 272, 1092, 3080, 3430, 3080, 1092, 272, 19, 23, 342, 1836, 5460, 9702, 9702, 5460, 1836, 342, 23

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Aug 23 2008

Keywords

• nonn
• easy
• tabl

Comments

For the purpose of this sequence define prime(0)=1.

Examples

			Triangle begins as:
   1;
   2,   2;
   3,   8,    3;
   5,  18,   18,     5;
   7,  40,   54,    40,     7;
  11,  70,  150,   150,    70,    11;
  13, 132,  315,   500,   315,   132,    13;
  17, 182,  693,  1225,  1225,   693,   182,    17;
  19, 272, 1092,  3080,  3430,  3080,  1092,   272,   19;
  23, 342, 1836,  5460,  9702,  9702,  5460,  1836,  342,  23;
  29, 460, 2565, 10200, 19110, 30492, 19110, 10200, 2565, 460, 29;
  ...
		

Links

• G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Crossrefs

Cf. A008578, A098350.

Programs

Magma

function A141617(n,k)
  if n eq 0 then return 1;
  else return Binomial(n,k)*NthPrime(k)*NthPrime(n-k);
  end if;
end function;
[A141617(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 26 2024

Maple

p:= n-> `if`(n=0, 1, ithprime(n)):
T:= (n, k)-> binomial(n, k)*p(k)*p(n-k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Apr 26 2023

Mathematica

A141617[n_, k_]:= If[n==0, 1, If[k==0 || k==n, Prime[n], Binomial[n, k]*Prime[k]*Prime[n-k]]];
Table[A414617[n,k], {n,0,12}, {k,0,n}]//Flatten

SageMath

def A141617(n,k):
    if n==0: return 1
    elif k==0 or k==n: return nth_prime(n)
    else: return binomial(n,k)*nth_prime(k)*nth_prime(n-k)
flatten([[A141617(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 26 2024

Formula

Symmetry: T(n, k) = T(n, n-k).