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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 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-François 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.)