A153521 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + 11*T(n-2, k-1), read by rows.

2, 11, 11, 2, 238, 2, 2, 1329, 1329, 2, 2, 1353, 5276, 1353, 2, 2, 1377, 21248, 21248, 1377, 2, 2, 1401, 37508, 100532, 37508, 1401, 2, 2, 1425, 54056, 371768, 371768, 54056, 1425, 2, 2, 1449, 70892, 838412, 1849388, 838412, 70892, 1449, 2, 2, 1473, 88016, 1503920, 6777248, 6777248, 1503920, 88016, 1473, 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) + 11*T(n-2, k-1).

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) = (0,1,5).

Sum_{k=1..n} T(n,k,p,q,j) = 2*(prime(j)-3)*[n=1] -2*prime(j)*(prime(j)-3)*[n=2] +2*prime(j)^2*(i*sqrt(prime(j)))^(n-3)*(ChebyshevU(n-3, -i/Sqrt(prime(j))) -((prime(j) -2)*i/sqrt(prime(j)))*ChebyshevU(n-4, -i/sqrt(prime(j)))) for (p,q,j)=(0,1,5) = A151617(n).

Row sums satisfy the recurrence relation S(n) = 2*S(n-1) + prime(j)*S(n-2), for n > 4, with S(1) = 2, S(2) = 2*prime(j), S(3) = 2*prime(j)^2, S(4) = 2*prime(j)^3 for j=5. (End)

Example

Triangle begins as:
   2;
  11,   11;
   2,  238,     2;
   2, 1329,  1329,       2;
   2, 1353,  5276,    1353,       2;
   2, 1377, 21248,   21248,    1377,       2;
   2, 1401, 37508,  100532,   37508,    1401,       2;
   2, 1425, 54056,  371768,  371768,   54056,    1425,     2;
   2, 1449, 70892,  838412, 1849388,  838412,   70892,  1449,    2;
   2, 1473, 88016, 1503920, 6777248, 6777248, 1503920, 88016, 1473, 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, 0, 1, 5], {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, 0, 1, 5) 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, 0, 1, 5): 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), this sequence (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), A153656 (2,3,9), A153657 (2,7,10).

Cf. A151617 (row sums).

Sequence in context: A064743 A376606 A109868 * A153650 A338049 A256665

Adjacent sequences: A153518 A153519 A153520 * A153522 A153523 A153524

Keyword

nonn,tabl,easy,less

Author

_Roger L. Bagula_, Dec 28 2008

Extensions

Edited by _G. C. Greubel_, Mar 04 2021

Status

approved