A153656 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (2*j +3)*prime(j)*T(n-2, k-1) with j=9, read by rows.

2, 23, 23, 2, 1054, 2, 2, 12165, 12165, 2, 2, 13133, 533412, 13133, 2, 2, 14101, 6422240, 6422240, 14101, 2, 2, 15069, 12779580, 270482476, 12779580, 15069, 2, 2, 16037, 19605432, 3385203976, 3385203976, 19605432, 16037, 2, 2, 17005, 26899796, 9577346548, 137413443860, 9577346548, 26899796, 17005, 2

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) + (2*j +3)*prime(j)*T(n-2, k-1) with j=9.

From G. C. Greubel, Mar 06 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) = (2,3,9).

Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1), for (p,q,j)=(2,3,9), = 2*A009967(n-1). (End)

Example

Triangle begins as:
   2;
  23,    23;
   2,  1054,        2;
   2, 12165,    12165,          2;
   2, 13133,   533412,      13133,            2;
   2, 14101,  6422240,    6422240,        14101,          2;
   2, 15069, 12779580,  270482476,     12779580,      15069,        2;
   2, 16037, 19605432, 3385203976,   3385203976,   19605432,    16037,     2;
   2, 17005, 26899796, 9577346548, 137413443860, 9577346548, 26899796, 17005, 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, 2, 3, 9], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 06 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, 2, 3, 9) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 06 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, 2, 3, 9): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 06 2021

Crossrefs

Sequences with variable (p,q,j): A153516 (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), this sequence (2,3,9), A153657 (2,7,10).

Cf. A009967 (powers of 23).

Sequence in context: A128365 A326954 A153654 * A242037 A233692 A084323

Adjacent sequences: A153653 A153654 A153655 * A153657 A153658 A153659

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Dec 30 2008

Extensions

Edited by G. C. Greubel, Mar 06 2021

Status

approved