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A293172 Triangle read by rows: T(n,k) = number of colored weighted Motzkin paths ending at (n,k).

%I A293172 #16 Apr 07 2020 08:47:45
%S A293172 1,6,1,40,10,1,280,84,14,1,2016,672,144,18,1,14784,5280,1320,220,22,1,
%T A293172 109824,41184,11440,2288,312,26,1,823680,320320,96096,21840,3640,420,
%U A293172 30,1,6223360,2489344,792064,198016,38080,5440,544,34,1,47297536,19348992,6449664,1736448,372096,62016,7752
%N A293172 Triangle read by rows: T(n,k) = number of colored weighted Motzkin paths ending at (n,k).
%H A293172 Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, <a href="https://doi.org/10.1016/j.disc.2017.07.006">Some matrix identities on colored Motzkin paths</a>, Discrete Mathematics 340.12 (2017): 3081-3091. See p. 3088.
%e A293172 Triangle begins:
%e A293172 1,
%e A293172 6,1,
%e A293172 40,10,1,
%e A293172 280,84,14,1,
%e A293172 2016,672,144,18,1,
%e A293172 14784,5280,1320,220,22,1,
%e A293172 ...
%p A293172 A293172 := proc(n,k)
%p A293172     option remember;
%p A293172     local b,d,r,c,e;
%p A293172     b := 4; d:= 2; r := 2 ; c := r^2 ; e := d ;
%p A293172     if k < 0 or k > n then
%p A293172         0;
%p A293172     elif k = n then
%p A293172         1;
%p A293172     elif k = 0 then
%p A293172         (b+e)*procname(n-1,0)+c*procname(n-1,1) ;
%p A293172     else
%p A293172         procname(n-1,k-1)+b*procname(n-1,k)+c*procname(n-1,k+1) ;
%p A293172     end if;
%p A293172 end proc:
%p A293172 seq(seq( A293172(n,k),k=0..n),n=0..15) ; # _R. J. Mathar_, Oct 27 2017
%t A293172 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]]]]];
%t A293172 Table[T[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 07 2020, from Maple *)
%Y A293172 First column is A069720.
%K A293172 nonn,tabl
%O A293172 0,2
%A A293172 _N. J. A. Sloane_, Oct 19 2017