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A293172 Triangle read by rows: T(n,k) = number of colored weighted Motzkin paths ending at (n,k). 1
1, 6, 1, 40, 10, 1, 280, 84, 14, 1, 2016, 672, 144, 18, 1, 14784, 5280, 1320, 220, 22, 1, 109824, 41184, 11440, 2288, 312, 26, 1, 823680, 320320, 96096, 21840, 3640, 420, 30, 1, 6223360, 2489344, 792064, 198016, 38080, 5440, 544, 34, 1, 47297536, 19348992, 6449664, 1736448, 372096, 62016, 7752 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..51.

Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091. See p. 3088.

EXAMPLE

Triangle begins:

1,

6,1,

40,10,1,

280,84,14,1,

2016,672,144,18,1,

14784,5280,1320,220,22,1,

...

MAPLE

A293172 := proc(n, k)

    option remember;

    local b, d, r, c, e;

    b := 4; d:= 2; r := 2 ; c := r^2 ; e := d ;

    if k < 0 or k > n then

        0;

    elif k = n then

        1;

    elif k = 0 then

        (b+e)*procname(n-1, 0)+c*procname(n-1, 1) ;

    else

        procname(n-1, k-1)+b*procname(n-1, k)+c*procname(n-1, k+1) ;

    end if;

end proc:

seq(seq( A293172(n, k), k=0..n), n=0..15) ; # R. J. Mathar, Oct 27 2017

MATHEMATICA

T[n_, k_] := T[n, k] = Module[{b = 4, d = 2, r = 2, c, e}, c = r^2; e = d; If[k < 0 || k > n, 0, If[k == n, 1, If[k == 0, (b + e) T[n - 1, 0] + c T[n - 1, 1], T[n - 1, k - 1] + b T[n - 1, k] + c T[n - 1, k + 1]]]]];

Table[T[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Apr 07 2020, from Maple *)

CROSSREFS

First column is A069720.

Sequence in context: A089504 A145927 A113365 * A145356 A145357 A035529

Adjacent sequences:  A293169 A293170 A293171 * A293173 A293174 A293175

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Oct 19 2017

STATUS

approved

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Last modified September 22 03:54 EDT 2020. Contains 337289 sequences. (Running on oeis4.)