OFFSET
1,1
LINKS
G. C. Greubel, Rows n = 1..50 of the triangle, flattened
FORMULA
T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j), T(3, 2, j) = 2*prime(j)^2 - 4, T(4, 2, j) = T(4, 3, j) = prime(j)^2 - 2, T(n, 1, j) = T(n, n, j) = 2 and j = 8.
Sum_{k=0..n} T(n, k, j) = 2*prime(j)^(n-1) for j=8 = 2*A001029(n-1).
EXAMPLE
Triangle begins as:
2;
19, 19;
2, 718, 2;
2, 6857, 6857, 2;
2, 7505, 245628, 7505, 2;
2, 8153, 2467944, 2467944, 8153, 2;
2, 8801, 4900212, 84273732, 4900212, 8801, 2;
2, 9449, 7542432, 886319856, 886319856, 7542432, 9449, 2;
2, 10097, 10394604, 2476630764, 28993055148, 2476630764, 10394604, 10097, 2;
MATHEMATICA
T[n_, k_, j_]:= T[n, k, 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, j] + T[n-1, k-1, j] + (2*j+1)*Prime[j]*T[n-2, k-1, j] ]]];
Table[T[n, k, 8], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 03 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, 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, j) + T(n-1, k-1, j) + (2*j+1)*nth_prime(j)*T(n-2, k-1, j)
flatten([[T(n, k, 8) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 03 2021
(Magma)
f:= func< n, j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
function T(n, k, 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, j) + T(n-1, k-1, j) + (2*j+1)*NthPrime(j)*T(n-2, k-1, j);
end if; return T;
end function;
[T(n, k, 8): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 03 2021
CROSSREFS
Cf. A001029 (powers of 19).
KEYWORD
AUTHOR
Roger L. Bagula, Dec 30 2008
EXTENSIONS
Edited by G. C. Greubel, Mar 03 2021
STATUS
approved