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

1,1

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 = 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)

Example

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;

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 *)

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

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.

Sequence in context: A175932 A370322 A225544 * A153657 A140152 A089536

Adjacent sequences: A153652 A153653 A153654 * A153656 A153657 A153658

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Dec 30 2008

Extensions

Edited by G. C. Greubel, Mar 03 2021

Status

approved