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

1,5

Comments

This triangle is dominated by the Eulerian numbers A008292.

Links

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

Formula

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

Example

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;

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A082680 A056939 A202924 * A174669 A140711 A164366

Adjacent sequences: A142592 A142593 A142594 * A142596 A142597 A142598

Keyword

nonn,easy,tabl

Author

Roger L. Bagula, Sep 22 2008

Extensions

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

Status

approved