A026374 Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= n-1.

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 6, 11, 6, 1, 1, 7, 17, 17, 7, 1, 1, 9, 30, 45, 30, 9, 1, 1, 10, 39, 75, 75, 39, 10, 1, 1, 12, 58, 144, 195, 144, 58, 12, 1, 1, 13, 70, 202, 339, 339, 202, 70, 13, 1, 1, 15, 95, 330, 685, 873, 685, 330, 95, 15, 1

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl

Comments

T(n,k) is number of lattice paths from (0,0) to (n,n-2k) using steps U=(1,1), D=(1,-1) and, at levels ...,-4,-2,0,2,4,..., also H=(2,0). Example: T(4,1)=6 because we have the following paths from (0,0) to (4,2): UUUD, UUH, UUDU, UDUU, HUU and DUUU. Row sums yield A026383. Column 1 is A032766, column 2 is A026381, column 3 is A026382. - Emeric Deutsch, Jan 25 2004

Examples

			Triangle starts:
  1;
  1,  1;
  1,  3,   1;
  1,  4,   4,   1;
  1,  6,  11,   6,    1;
  1,  7,  17,  17,    7,    1;
  1,  9,  30,  45,   30,    9,    1;
  1, 10,  39,  75,   75,   39,   10,    1;
  1, 12,  58, 144,  195,  144,   58,   12,   1;
  1, 13,  70, 202,  339,  339,  202,   70,  13,   1;
  1, 15,  95, 330,  685,  873,  685,  330,  95,  15,  1;
  1, 16, 110, 425, 1015, 1558, 1558, 1015, 425, 110, 16, 1;
		

Links

• Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
• Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

Crossrefs

Cf. A026383, A051159,A169623, A007318

Cf. A026375 (central terms).

Programs

Haskell

a026374 n k = a026374_tabl !! n !! k
a026374_row n = a026374_tabl !! n
a026374_tabl = [1] : map fst (map snd $ iterate f (1, ([1, 1], [1]))) where
   f (0, (us, vs)) = (1, (zipWith (+) ([0] ++ us) (us ++ [0]), us))
   f (1, (us, vs)) = (0, (zipWith (+) ([0] ++ vs ++ [0]) $
                             zipWith (+) ([0] ++ us) (us ++ [0]), us))
-- Reinhard Zumkeller, Feb 22 2014

Mathematica

p[x, 1] := 1;
p[x_, n_] := p[x, n] = If[Mod[n, 2] == 0, (x + 1)*p[x, n - 1], (x^2 + 1)^Floor[n/2]];
a = Table[CoefficientList[p[x, n], x], {n, 1, 12}];
Flatten[a] (* Roger L. Bagula and Gary W. Adamson, Dec 04 2009 *)

Formula

T(n, k) = number of integer strings s(0), ..., s(n) such that s(0)=0, s(n) = n-2k, where, for 1 <= i <= n, s(i) is even if i is even and |s(i) - s(i-1)| <= 1.

From Emeric Deutsch, Jan 25 2004: (Start)

T(2n, k) = Sum_{j=ceiling(k/2)..k} 3^(2j-k)*binomial(n, j)*binomial(j, k-j);

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

G.f.: (1 + z + t*z)/(1 - (1+3*t+t^2)*z^2) = 1 + (1+t)*z + (1+3*t+t^2)*z^2+ ... .

Generating polynomial for row 2n is (1 + 3*t + t^2)^n;

Generating polynomial for row 2n+1 it is (1+t)*(1 + 3*t + t^2)^n. (End)

From Emeric Deutsch, Jan 30 2004: (Start)

T(2n, k) = Sum_{j=ceiling(k/2)..k} 3^(2j-k)*binomial(n, j)*binomial(j, k-j);

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