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A153653
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 = 8, read by rows.
14
2, 19, 19, 2, 718, 2, 2, 6857, 6857, 2, 2, 7505, 245628, 7505, 2, 2, 8153, 2467944, 2467944, 8153, 2, 2, 8801, 4900212, 84273732, 4900212, 8801, 2, 2, 9449, 7542432, 886319856, 886319856, 7542432, 9449, 2, 2, 10097, 10394604, 2476630764, 28993055148, 2476630764, 10394604, 10097, 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 = 8.
Sum_{k=0..n} T(n, k, j) = 2*prime(j)^(n-1) for j=8 = 2*A001029(n-1).
EXAMPLE
Triangle begins as:
2;
19, 19;
2, 718, 2;
2, 6857, 6857, 2;
2, 7505, 245628, 7505, 2;
2, 8153, 2467944, 2467944, 8153, 2;
2, 8801, 4900212, 84273732, 4900212, 8801, 2;
2, 9449, 7542432, 886319856, 886319856, 7542432, 9449, 2;
2, 10097, 10394604, 2476630764, 28993055148, 2476630764, 10394604, 10097, 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, 8], {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, 8) 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, 8): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 03 2021
CROSSREFS
Cf. A153652 (j=7), this sequence (j=8), A153654 (j=9), A153655 (j=10).
Cf. A001029 (powers of 19).
Sequence in context: A335363 A176618 A356477 * A065643 A038031 A229264
KEYWORD
nonn,tabl,easy,less
AUTHOR
Roger L. Bagula, Dec 30 2008
EXTENSIONS
Edited by G. C. Greubel, Mar 03 2021
STATUS
approved