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) + 7*T(n-2, k-1).
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) = (0,1,4).
Sum_{k=1..n} T(n,k,p,q,j) = 2*(prime(j)-3)*[n=1] -2*prime(j)*(prime(j)-3)*[n=2] +2*prime(j)^2*(i*sqrt(prime(j)))^(n-3)*(ChebyshevU(n-3, -i/Sqrt(prime(j))) -((prime(j) -2)*i/sqrt(prime(j)))*ChebyshevU(n-4, -i/sqrt(prime(j)))) for (p,q,j)=(0,1,4).
Row sums satisfy the recurrence relation S(n) = 2*S(n-1) + prime(j)*S(n-2), for n > 4, with S(1) = 2, S(2) = 2*prime(j), S(3) = 2*prime(j)^2, S(4) = 2*prime(j)^3 with j=4. (End)
EXAMPLE
Triangle begins as:
2;
7, 7;
2, 94, 2;
2, 341, 341, 2;
2, 357, 1340, 357, 2;
2, 373, 4084, 4084, 373, 2;
2, 389, 6956, 17548, 6956, 389, 2;
2, 405, 9956, 53092, 53092, 9956, 405, 2;
2, 421, 13084, 111740, 229020, 111740, 13084, 421, 2;
2, 437, 16340, 194516, 712404, 712404, 194516, 16340, 437, 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, 0, 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, 0, 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, 0, 1, 4): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 04 2021
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Dec 28 2008
EXTENSIONS
Edited by G. C. Greubel, Mar 04 2021
STATUS
approved