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

Author

Roger L. Bagula, Sep 22 2008

Keywords

• nonn
• tabl

Examples

			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;
		

Links

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

Crossrefs

Cf. A008292, A047851, A060187, A119258.

Programs

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

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 *)

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

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

Extensions

Edited by G. C. Greubel, Apr 13 2021