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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
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-François Alcover, Apr 07 2020, from Maple *)
CROSSREFS
First column is A069720.
Sequence in context: A089504 A145927 A113365 * A145356 A145357 A035529
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Oct 19 2017
STATUS
approved

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Last modified June 24 16:36 EDT 2024. Contains 373680 sequences. (Running on oeis4.)