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A153652
Triangle 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) and j = 7, read by rows.
14
2, 17, 17, 2, 574, 2, 2, 4911, 4911, 2, 2, 5423, 156192, 5423, 2, 2, 5935, 1413920, 1413920, 5935, 2, 2, 6447, 2802720, 42656800, 2802720, 6447, 2, 2, 6959, 4322592, 406009120, 406009120, 4322592, 6959, 2, 2, 7471, 5973536, 1125025312, 11689502240, 1125025312, 5973536, 7471, 2
OFFSET
1,1
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 = 7.
Sum_{k=0..n} T(n, k, j) = 2*prime(j)^(n-1) for j=7 = 2*A001026(n-1).
EXAMPLE
Triangle begins as:
2;
17, 17;
2, 574, 2;
2, 4911, 4911, 2;
2, 5423, 156192, 5423, 2;
2, 5935, 1413920, 1413920, 5935, 2;
2, 6447, 2802720, 42656800, 2802720, 6447, 2;
2, 6959, 4322592, 406009120, 406009120, 4322592, 6959, 2;
2, 7471, 5973536, 1125025312, 11689502240, 1125025312, 5973536, 7471, 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, 7], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 02 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, 7) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 02 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, 7): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 02 2021
CROSSREFS
Cf. this sequence (j=7), A153653 (j=8), A153654 (j=9), A153655 (j=10).
Cf. A001026 (powers of 17).
Sequence in context: A077311 A196732 A346391 * A198290 A198596 A198408
KEYWORD
nonn,tabl,easy,less
AUTHOR
Roger L. Bagula, Dec 30 2008
EXTENSIONS
Edited by G. C. Greubel, Mar 02 2021
STATUS
approved