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.

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

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

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. A153516, A153518, A153520, A153521, A153648, A153649, A153650, A153651, A153656, A153657.

Cf. A001029 (powers of 19).

Sequence in context: A335363 A176618 A356477 * A065643 A038031 A229264

Adjacent sequences: A153650 A153651 A153652 * A153654 A153655 A153656

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Dec 30 2008

Extensions

Edited by G. C. Greubel, Mar 03 2021

Status

approved