A142596 Triangle T(n, k) = T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k-1), with T(n,1) = T(n, n) = 1, read by rows.

1, 1, 1, 1, 6, 1, 1, 21, 21, 1, 1, 66, 126, 66, 1, 1, 201, 576, 576, 201, 1, 1, 606, 2331, 3456, 2331, 606, 1, 1, 1821, 8811, 17361, 17361, 8811, 1821, 1, 1, 5466, 31896, 78516, 104166, 78516, 31896, 5466, 1, 1, 16401, 112086, 331236, 548046, 548046, 331236, 112086, 16401, 1

Offset

1,5

Links

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

Formula

T(n, k) = T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k-1), with T(n,1) = T(n, n) = 1.

Sum_{k=1..n} T(n, k) = (6^(n-1) + 4)/5 = A047851(n-1). - G. C. Greubel, Apr 13 2021

Example

The triangle begins as:
  1;
  1,     1;
  1,     6,      1;
  1,    21,     21,      1;
  1,    66,    126,     66,      1;
  1,   201,    576,    576,    201,      1;
  1,   606,   2331,   3456,   2331,    606,      1;
  1,  1821,   8811,  17361,  17361,   8811,   1821,      1;
  1,  5466,  31896,  78516, 104166,  78516,  31896,   5466,     1;
  1, 16401, 112086, 331236, 548046, 548046, 331236, 112086, 16401, 1;

Mathematica

T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, T[n-1, k-1] +3*T[n-1, k] +2*T[n-1, k-1]];
Table[T[n, k], {n, 10}, {k, n}]//Flatten (* modified by G. C. Greubel, Apr 13 2021 *)

Prog

(Magma)
function T(n, k)
  if k eq 1 or k eq n then return 1;
  else return T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k-1);
  end if; return T;
end function;
[T(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 13 2021
(Sage)
@CachedFunction
def T(n, k): return 1 if k==1 or k==n else T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k-1)
flatten([[T(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 13 2021

Crossrefs

Cf. A008292, A047851, A060187, A119258.

Sequence in context: A056941 A157638 A347975 * A176063 A350060 A155467

Adjacent sequences: A142593 A142594 A142595 * A142597 A142598 A142599

Keyword

nonn,tabl

Author

Roger L. Bagula, Sep 22 2008

Extensions

Edited by G. C. Greubel, Apr 13 2021

Status

approved