A142595 Triangle T(n,k) = 2*T(n-1, k-1) + 2*T(n-1, k), read by rows.

1, 1, 1, 1, 4, 1, 1, 10, 10, 1, 1, 22, 40, 22, 1, 1, 46, 124, 124, 46, 1, 1, 94, 340, 496, 340, 94, 1, 1, 190, 868, 1672, 1672, 868, 190, 1, 1, 382, 2116, 5080, 6688, 5080, 2116, 382, 1, 1, 766, 4996, 14392, 23536, 23536, 14392, 4996, 766, 1

Offset: 1

Author

Roger L. Bagula, Sep 22 2008

Keywords

• nonn
• easy
• tabl

Comments

This triangle is dominated by the Eulerian numbers A008292.

Examples

			Triangle begins as:
  1;
  1,   1;
  1,   4,    1;
  1,  10,   10,     1;
  1,  22,   40,    22,     1;
  1,  46,  124,   124,    46,     1;
  1,  94,  340,   496,   340,    94,     1;
  1, 190,  868,  1672,  1672,   868,   190,    1;
  1, 382, 2116,  5080,  6688,  5080,  2116,  382,   1;
  1, 766, 4996, 14392, 23536, 23536, 14392, 4996, 766, 1;
		

Links

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

Crossrefs

Cf. A008292, A047849 (row sums), A119258.

Programs

Magma

function T(n,k)
  if k eq 1 or k eq n then return 1;
  else return 2*(T(n-1, k-1) + T(n-1, k));
  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, 2*(T[n-1, k-1] +T[n-1, k])];
Table[T[n, k], {n, 10}, {k, n}]//Flatten (* modified by G. C. Greubel, Apr 13 2021 *)
a[0] = {1}; a[1] = {1, 1};
a[n_]:= a[n]= 2*Join[a[n-1], {-1/2}] + 2*Join[{-1/2}, a[n-1]];
Table[a[n], {n,0,10}]//Flatten (* Roger L. Bagula, Dec 09 2008 *)

Sage

@CachedFunction
def T(n,k): return 1 if k==1 or k==n else 2*(T(n-1, k-1) + T(n-1, k))
flatten([[T(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 13 2021

Formula

Sum_{k=0..n} T(n, k) = (4^(n-1) + 2)/3 = A047849(n-1).

Extensions

Edited by N. J. A. Sloane, Dec 11 2008