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A293171 Triangle read by rows: T(n,k) = number of colored weighted Motzkin paths ending at (n,k). 1
1, 1, 1, 9, 2, 1, 25, 15, 3, 1, 145, 52, 22, 4, 1, 561, 285, 90, 30, 5, 1, 2841, 1206, 495, 140, 39, 6, 1, 12489, 6027, 2261, 791, 203, 49, 7, 1, 60705, 27560, 11452, 3864, 1190, 280, 60, 8, 1, 281185, 134073, 54468, 20076, 6174, 1710, 372, 72, 9, 1, 1353769, 633130, 268845, 99240, 33090, 9372, 2370, 480, 85, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Table of n, a(n) for n=0..65.

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. 3087.

EXAMPLE

Triangle begins:

1,

1,1,

9,2,1,

25,15,3,1,

145,52,22,4,1,

561,285,90,30,5,1,

...

MAPLE

A293171 := proc(n, k)

    option remember;

    local b, e, c;

    b := 1; e:= 2; c := e^2 ;

    if k < 0 or k > n then

        0;

    elif k = n then

        1;

    elif k = 0 then

        b*procname(n-1, 0)+2*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( A293171(n, k), k=0..n), n=0..15) ; # R. J. Mathar, Oct 27 2017

MATHEMATICA

T[n_, k_] := T[n, k] = Module[{b=1, e=2, c=4}, Which[k<0 || k>n, 0, k==n, 1, k == 0, b*T[n-1, 0] + 2*c*T[n-1, 1], True, T[n-1, k-1] + b*T[n-1, k] + c*T[n-1, k+1]]];

Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Dec 19 2019, after R. J. Mathar *)

CROSSREFS

First column is A084605, 2nd A098520.

Sequence in context: A293258 A010536 A239908 * A334689 A151898 A080994

Adjacent sequences:  A293168 A293169 A293170 * A293172 A293173 A293174

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Oct 19 2017

STATUS

approved

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Last modified September 29 17:04 EDT 2020. Contains 337432 sequences. (Running on oeis4.)