A153655 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 = 10, read by rows.

2, 29, 29, 2, 1678, 2, 2, 24387, 24387, 2, 2, 25607, 1070676, 25607, 2, 2, 26827, 15947966, 15947966, 26827, 2, 2, 28047, 31569456, 683937616, 31569456, 28047, 2, 2, 29267, 47935146, 10427818366, 10427818366, 47935146, 29267, 2, 2, 30487, 65045036, 29701552216, 437373644876, 29701552216, 65045036, 30487, 2

Offset: 1

Author

Roger L. Bagula, Dec 30 2008

Keywords

• nonn
• tabl
• easy
• less

Examples

			Triangle begins as:
   2;
  29,    29;
   2,  1678,        2;
   2, 24387,    24387,           2;
   2, 25607,  1070676,       25607,            2;
   2, 26827, 15947966,    15947966,        26827,           2;
   2, 28047, 31569456,   683937616,     31569456,       28047,        2;
   2, 29267, 47935146, 10427818366,  10427818366,    47935146,    29267,     2;
   2, 30487, 65045036, 29701552216, 437373644876, 29701552216, 65045036, 30487, 2;
		

Links

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

Crossrefs

Cf. A153652 (j=7), A153653 (j=8), A153654 (j=9), this sequence (j=10).

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

Programs

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,10): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 03 2021

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,10], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 03 2021 *)

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,10) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 03 2021

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 = 10.

From G. C. Greubel, Mar 04 2021: (Start)

Sum_{k=0..n} T(n, k, 10) = -(76/147)*[n=0] - (116/7)*[n=1] + 1682*(i*sqrt(609))^(n-2)*(ChebyshevU(n-2, -i/sqrt(609)) - (27*i/sqrt(609))*ChebyshevU(n-3, -i/sqrt(609) )).

Row sums satisfy the recurrence S(n) = 2*S(n-1) + 609*S(n-2) for n>4 with S(0) = 2, S(1) = 58, S(2) = 1682, S(3) = 48778. (End)

Extensions

Edited by G. C. Greubel, Mar 03 2021