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 = 9.
From G. C. Greubel, Mar 04 2021: (Start)
Sum_{k=0..n} T(n, k, 10) = -(68/361)*[n=0] - (92/19)*[n=1] + 1058*(i*sqrt(437))^(n-2)*(ChebyshevU(n-2, -i/sqrt(437)) - (21*i/sqrt(437))*ChebyshevU(n-3, -i/sqrt(437) )).
Row sums satisfy the recurrence S(n) = 2*S(n-1) + 437*S(n-2) for n>4 with S(0) = 2, S(1) = 46, S(2) = 1058, S(3) = 24334. (End)
EXAMPLE
Triangle begins as:
2;
23, 23;
2, 1054, 2;
2, 12165, 12165, 2;
2, 13041, 484928, 13041, 2;
2, 13917, 5814074, 5814074, 13917, 2;
2, 14793, 11526908, 223541684, 11526908, 14793, 2;
2, 15669, 17623430, 2775818930, 2775818930, 17623430, 15669, 2;
2, 16545, 24103640, 7830701156, 103239353768, 7830701156, 24103640, 16545, 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, 9], {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, 9) 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, 9): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 03 2021
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Dec 30 2008
EXTENSIONS
Edited by G. C. Greubel, Mar 03 2021
STATUS
approved