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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. 18
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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

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.

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.

T(2n, k)=sum(3^(2j-k)*binomial(n, j)binomial(j, k-j), j=ceil(k/2)..k); T(2n+1, k)=T(2n, k-1)+T(2n, k). G.f.=(1+z+tz)/[1-(1+3t+t^2)z^2]=1+(1+t)z+(1+3t+t^2)z^2+... . Generating polynomial for row 2n is (1+3t+t^2)^n and for row 2n+1 it is (1+t)(1+3t+t^2)^n. - Emeric Deutsch, Jan 25 2004

T(2n, k)=sum(3^(2j-k)*binomial(n, j)*binomial(j, k-j), j=ceil(k/2)..k); T(2n+1, k)=T(2n, k-1)+T(2n, k). - Emeric Deutsch, Jan 30 2004

w=3:\q p(x,n)=If[Mod[n, 2] == 0, (x + 1)*p(x, n - 1), (x^2 + w*x + 1)^Floor[n/2]. [Roger L. Bagula and Gary W. Adamson, Dec 04 2009]

EXAMPLE

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} (End)

MATHEMATICA

w = 0;

p[x, 1] := 1;

p[x_, n_] := p[x, n] = If[Mod[n, 2] == 0, (x + 1)*p[x, n - 1], (x^2 + w*x + 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 *)

PROG

(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

CROSSREFS

Cf. A026383, A051159,A169623, A007318

Cf. A026375 (central terms).

Sequence in context: A136482 A026648 A026747 * A174032 A180979 A102716

Adjacent sequences:  A026371 A026372 A026373 * A026375 A026376 A026377

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified February 20 00:28 EST 2019. Contains 320329 sequences. (Running on oeis4.)