A028262 Elements in 3-Pascal triangle (by row).

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 5, 8, 5, 1, 1, 6, 13, 13, 6, 1, 1, 7, 19, 26, 19, 7, 1, 1, 8, 26, 45, 45, 26, 8, 1, 1, 9, 34, 71, 90, 71, 34, 9, 1, 1, 10, 43, 105, 161, 161, 105, 43, 10, 1, 1, 11, 53, 148, 266, 322, 266, 148, 53, 11, 1, 1, 12, 64, 201, 414, 588, 588, 414, 201, 64, 12, 1

Offset: 0

Author

Mohammad K. Azarian

Keywords

• nonn
• nice
• tabl

Examples

			Triangle begins:
  1;
  1 1;
  1 3 1;
  1 4 4 1;
  1 5 8 5 1;
  ...
		

Links

• Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
• László Németh, Tetrahedron trinomial coefficient transform, Integers (2019) Vol. 19, Article A41.
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A007318, A028275, A072405.

Programs

Haskell

a028262 n k = a028262_tabl !! n !! k
a028262_row n = a028262_tabl !! n
a028262_tabl = [1] : [1,1] : iterate
   (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [1,3,1]
-- Reinhard Zumkeller, Aug 02 2012

Magma

T:= func< n,k | n lt 2 select 1 else Binomial(n, k) + Binomial(n-2, k-1) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 28 2021

Mathematica

T[n_, k_]:= If[n==1, 1, Binomial[n, k] + Binomial[n-2, k-1]]; Table[T[n, k], {n, 0, 11}, {k, 0, n}]//Flatten (* Jean-François Alcover, Jan 28 2015 *)

Sage

def T(n,k): return 1 if n<2 else binomial(n,k) + binomial(n-2,k-1)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 28 2021

Formula

After the 3rd row, use Pascal's rule.

From Ralf Stephan, Jan 31 2005: (Start)

T(n, k) = C(n, k) + C(n-2, k-1).

G.f.: (1 + x^2*y)/(1 - x*(1+y)). (End)

T(n+2,k+1) = A007318(n,k) - A007318(n+2,k+1); 0 < k < n. - Reinhard Zumkeller, Aug 02 2012

Sum_{k=0..n} T(n,k) = (n+1)*[n<2] + 5*2^(n-2)*[n>=2]. - G. C. Greubel, Apr 28 2021

Extensions

More terms from James Sellers