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A153516
Triangle 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) and (p,q,j) = (0,1,2), read by rows.
14
2, 3, 3, 2, 14, 2, 2, 25, 25, 2, 2, 33, 92, 33, 2, 2, 41, 200, 200, 41, 2, 2, 49, 340, 676, 340, 49, 2, 2, 57, 512, 1616, 1616, 512, 57, 2, 2, 65, 716, 3148, 5260, 3148, 716, 65, 2, 2, 73, 952, 5400, 13256, 13256, 5400, 952, 73, 2
OFFSET
1,1
FORMULA
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) = (0,1,2).
Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1) for (p,q,j) = (0,1,2) = 2*A000244(n-1).
EXAMPLE
Triangle begins as:
2;
3, 3;
2, 14, 2;
2, 25, 25, 2;
2, 33, 92, 33, 2;
2, 41, 200, 200, 41, 2;
2, 49, 340, 676, 340, 49, 2;
2, 57, 512, 1616, 1616, 512, 57, 2;
2, 65, 716, 3148, 5260, 3148, 716, 65, 2;
2, 73, 952, 5400, 13256, 13256, 5400, 952, 73, 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, 2], {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, 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, 2) 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, 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, 2): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 03 2021
CROSSREFS
Sequences with variable (p,q,j): this sequence (0,1,2), A153518 (0,1,3), A153520 (0,1,4), A153521 (0,1,5), A153648 (1,0,3), A153649 (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. A000244.
Sequence in context: A175393 A338307 A153290 * A153311 A153312 A153283
KEYWORD
nonn,tabl,easy,less
AUTHOR
Roger L. Bagula, Dec 28 2008
EXTENSIONS
Edited by G. C. Greubel, Mar 03 2021
STATUS
approved