A124960 Triangle read by rows: T(n,k) = p(k)*T(n-1,k) + T(n-1,k-1) (1 <= k <= n), where p(k) denotes the k-th prime.

1, 2, 1, 4, 5, 1, 8, 19, 10, 1, 16, 65, 69, 17, 1, 32, 211, 410, 188, 28, 1, 64, 665, 2261, 1726, 496, 41, 1, 128, 2059, 11970, 14343, 7182, 1029, 58, 1, 256, 6305, 61909, 112371, 93345, 20559, 2015, 77, 1, 512, 19171, 315850, 848506, 1139166, 360612, 54814, 3478, 100, 1

Offset: 1

Author

Gary W. Adamson, Nov 13 2006

Keywords

• nonn
• tabl

Examples

			Triangle starts:
   1;
   2,   1;
   4,   5,   1;
   8,  19,  10,   1;
  16,  65,  69,  17,  1;
  32, 211, 410, 188, 28, 1;
		

Links

• G. C. Greubel, Rows n = 1..100 of triangle, flattened

Crossrefs

T(2n,n) gives A332967 (for n>0).

Programs

Magma

function T(n,k)
  if k lt 1 or k gt n then return 0;
  elif n eq 1 and k eq 1 then return 1;
  else return NthPrime(k)*T(n-1,k) + T(n-1,k-1);
  end if;
  return T;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 19 2019

Maple

T:=proc(n,k): if n=1 and k=1 then 1 elif k<1 or k>n then 0 else ithprime(k)*T(n-1,k)+T(n-1,k-1) fi end: for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

Mathematica

T[n_, k_]:= T[n, k]= If[n==1 && k==1 , 1, If[k<1 || k>n, 0, Prime[k]*T[n-1, k] + T[n-1, k-1] ]]; Table[T[n, k], {n,12}, {k, n}]//Flatten (* G. C. Greubel, Nov 19 2019 *)

PARI

T(n,k) = if(n==1 && k==1, 1, if(k<1 || k>n, 0, prime(k)*T(n-1, k) + T(n-1, k-1) )); \\ G. C. Greubel, Nov 19 2019

Sage

@CachedFunction
def T(n,k):
    if (k<1 or k>n): return 0
    elif (n==1 and k==1): return 1
    else: return nth_prime(k)*T(n-1, k) + T(n-1, k-1)
[[T(n,k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Nov 19 2019

Extensions

Edited by N. J. A. Sloane, Nov 29 2006