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) + 5*T(n-2, k-1).
Recurrence row sums: s(n) = 2*s(n-1) + 5*s(n-2), n > 4, with s(1) = 2, s(2) = 10, s(3) = 50, s(4) = 250. - R. J. Mathar, Jan 22 2009
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) = (0,1,3).
Sum_{k=0..n} T(n,k,0,1,3) = 4*(-5)^n*[n<2] + 50*(i*sqrt(5))^(n-2)*(ChebyshevU(n-2, -i/sqrt(5)) - (3*i/sqrt(5))*ChebyshevU(n-3, -i/sqrt(5))) = 4*(-5)^n*[n<2] + 50*A123011(n-2). (End)
EXAMPLE
Triangle begins as:
2;
5, 5;
2, 46, 2;
2, 123, 123, 2;
2, 135, 476, 135, 2;
2, 147, 1226, 1226, 147, 2;
2, 159, 2048, 4832, 2048, 159, 2;
2, 171, 2942, 13010, 13010, 2942, 171, 2;
2, 183, 3908, 26192, 50180, 26192, 3908, 183, 2;
2, 195, 4946, 44810, 141422, 141422, 44810, 4946, 195, 2;
MAPLE
A153518 := proc(n, k) option remember ; if n =1 then 2; elif n = 2 then 5; elif k=1 or k=n then 2; elif n = 3 then 46 ; elif n = 4 then 123 ; else procname(n-1, k-1)+procname(n-1, k)+5*procname(n-2, k-1) ; end: end: for n from 1 to 13 do for k from 1 to n do printf("%d, ", A153518(n, k)) ; od: od: # R. J. Mathar, Jan 22 2009
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, 3], {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, 3) 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, 3): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 04 2021
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Dec 28 2008
EXTENSIONS
More terms from R. J. Mathar, Jan 22 2009
Edited by G. C. Greubel, Mar 04 2021
STATUS
approved