A153649 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+1)*prime(j)*T(n-2, k-1) with j=4, read by rows.

2, 7, 7, 2, 94, 2, 2, 341, 341, 2, 2, 413, 3972, 413, 2, 2, 485, 16320, 16320, 485, 2, 2, 557, 31260, 171660, 31260, 557, 2, 2, 629, 48792, 774120, 774120, 48792, 629, 2, 2, 701, 68916, 1917012, 7556340, 1917012, 68916, 701, 2, 2, 773, 91632, 3693648, 36567552, 36567552, 3693648, 91632, 773, 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+1)*prime(j)*T(n-2, k-1) with j=4.

From G. C. Greubel, Mar 04 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) = (1,1,4).

Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1), for (p,q,j)=(1,1,4), = 2*A000420(n-1). (End)

Example

Triangle begins as:
  2;
  7,   7;
  2,  94,     2;
  2, 341,   341,       2;
  2, 413,  3972,     413,        2;
  2, 485, 16320,   16320,      485,        2;
  2, 557, 31260,  171660,    31260,      557,       2;
  2, 629, 48792,  774120,   774120,    48792,     629,     2;
  2, 701, 68916, 1917012,  7556340,  1917012,   68916,   701,   2;
  2, 773, 91632, 3693648, 36567552, 36567552, 3693648, 91632, 773, 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, 1, 4], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 04 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, 1, 4) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 04 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, 1, 4): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 04 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), this sequence (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), A153656 (2,3,9), A153657 (2,7,10).

Cf. A000420 (powers of 7).

Sequence in context: A374525 A374526 A153520 * A020770 A177003 A164767

Adjacent sequences: A153646 A153647 A153648 * A153650 A153651 A153652

Keyword

nonn,tabl

Author

Roger L. Bagula, Dec 30 2008

Extensions

Edited by G. C. Greubel, Mar 04 2021

Status

approved