OFFSET
1,1
LINKS
G. C. Greubel, Rows n = 1..50 of the triangle, flattened
FORMULA
T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+1)*prime(j)*T(n-2, k-1) with j=4.
From G. C. Greubel, Mar 04 2021: (Start)
T(n,k,p,q,j) = T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*prime(j)*T(n-2,k-1,p,q,j) with T(2,k,p,q,j) = prime(j), T(3,2,p,q,j) = 2*prime(j)^2 -4, T(4,2,p,q,j) = T(4,3,p,q,j) = prime(j)^2 -2, T(n,1,p,q,j) = T(n,n,p,q,j) = 2 and (p,q,j) = (1,1,4).
Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1), for (p,q,j)=(1,1,4), = 2*A000420(n-1). (End)
EXAMPLE
Triangle begins as:
2;
7, 7;
2, 94, 2;
2, 341, 341, 2;
2, 413, 3972, 413, 2;
2, 485, 16320, 16320, 485, 2;
2, 557, 31260, 171660, 31260, 557, 2;
2, 629, 48792, 774120, 774120, 48792, 629, 2;
2, 701, 68916, 1917012, 7556340, 1917012, 68916, 701, 2;
2, 773, 91632, 3693648, 36567552, 36567552, 3693648, 91632, 773, 2;
MATHEMATICA
T[n_, k_, p_, q_, j_]:= T[n, k, p, q, j]= If[n==2, Prime[j], If[n==3 && k==2 || n==4 && 2<=k<=3, ((3-(-1)^n)/2)*Prime[j]^(n-1) -2^((3-(-1)^n)/2), If[k==1 || k==n, 2, T[n-1, k, p, q, j] + T[n-1, k-1, p, q, j] + (p*j+q)*Prime[j]*T[n-2, k-1, p, q, j] ]]];
Table[T[n, k, 1, 1, 4], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 04 2021 *)
PROG
(Sage)
@CachedFunction
def f(n, j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)
def T(n, k, p, q, j):
if (n==2): return nth_prime(j)
elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n, j)
elif (k==1 or k==n): return 2
else: return T(n-1, k, p, q, j) + T(n-1, k-1, p, q, j) + (p*j+q)*nth_prime(j)*T(n-2, k-1, p, q, j)
flatten([[T(n, k, 1, 1, 4) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 04 2021
(Magma)
f:= func< n, j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
function T(n, k, p, q, j)
if n eq 2 then return NthPrime(j);
elif (n eq 3 and k eq 2 or n eq 4 and k eq 2 or n eq 4 and k eq 3) then return f(n, j);
elif (k eq 1 or k eq n) then return 2;
else return T(n-1, k, p, q, j) + T(n-1, k-1, p, q, j) + (p*j+q)*NthPrime(j)*T(n-2, k-1, p, q, j);
end if; return T;
end function;
[T(n, k, 1, 1, 4): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 04 2021
CROSSREFS
Sequences with variable (p,q,j): A153516 (0,1,2), A153518 (0,1,3), A153520 (0,1,4), A153521 (0,1,5), A153648 (1,0,3), this sequence (1,1,4), A153650 (1,4,5), A153651 (1,5,6), A153652 (2,1,7), A153653 (2,1,8), A153654 (2,1,9), A153655 (2,1,10), A153656 (2,3,9), A153657 (2,7,10).
Cf. A000420 (powers of 7).
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Dec 30 2008
EXTENSIONS
Edited by G. C. Greubel, Mar 04 2021
STATUS
approved