A153651 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+5)*prime(j)*T(n-2, k-1) with j=6, read by rows.

2, 13, 13, 2, 334, 2, 2, 2195, 2195, 2, 2, 2483, 52152, 2483, 2, 2, 2771, 368520, 368520, 2771, 2, 2, 3059, 726360, 8194776, 726360, 3059, 2, 2, 3347, 1125672, 61619496, 61619496, 1125672, 3347, 2, 2, 3635, 1566456, 166614648, 1295091960, 166614648, 1566456, 3635, 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) + (j+5)*prime(j)*T(n-2, k-1) with j=6.

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) = (1,5,6).

Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1) for j=6, = 2*A001022(n-1). (End)

Example

Triangle begins as:
   2;
  13,   13;
   2,  334,       2;
   2, 2195,    2195,         2;
   2, 2483,   52152,      2483,          2;
   2, 2771,  368520,    368520,       2771,         2;
   2, 3059,  726360,   8194776,     726360,      3059,       2;
   2, 3347, 1125672,  61619496,   61619496,   1125672,    3347,    2;
   2, 3635, 1566456, 166614648, 1295091960, 166614648, 1566456, 3635, 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, 1, 5, 6], {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, 1, 5, 6) 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, 1, 5, 6): 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), this sequence (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. A001022 (powers of 13).

Sequence in context: A002591 A037055 A065584 * A369411 A229908 A335972

Adjacent sequences: A153648 A153649 A153650 * A153652 A153653 A153654

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Dec 30 2008

Extensions

Edited by G. C. Greubel, Mar 06 2021

Status

approved