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%I #10 Nov 15 2019 23:37:02
%S 1,3,2,11,12,4,45,62,36,8,197,312,240,96,16,903,1570,1440,784,240,32,
%T 4279,7956,8244,5472,2320,576,64,20793,40670,46116,35224,18480,6432,
%U 1344,128,103049,209712,254912,216384,132320,57600,17024,3072,256
%N Triangle read by rows: T(n,k) is the number of paths of length n in the first quadrant, starting at the origin, ending at height k and consisting of 2 kind of upsteps U=(1,1) (U1 and U2), 3 kind of flatsteps F=(1,0) (F1, F2 and F3) and 1 kind of downsteps D=(1,-1).
%C T(n,0) = A001003(n+1) (the little Schroeder numbers).
%C Row sums yield A134425.
%F T(n,k) = ((k+1)*2^k/(n+1))*Sum_{j=0..n-k} binomial(n+1, j)*binomial(n+1, k+j+1)*2^j (0 <= k <= n).
%F G.f.: g/(1-2*t*z*g), where g = 1 + 3*z*g + 2*z^2*g^2 is the g.f. of the little Schroeder numbers 1, 3, 11, 45, 197, ... (A001003).
%e T(2,1)=12 because we have 6 paths of shape FU and 6 paths of shape UF.
%e Triangle starts:
%e 1;
%e 3, 2;
%e 11, 12, 4;
%e 45, 62, 36, 8;
%e 197, 312, 240, 96, 16;
%p T:=proc(n,k) options operator,arrow: 2^k*(k+1)*(sum(2^j*binomial(n+1,j)*binomial(n+1, k+1+j),j=0..n-k))/(n+1) end proc: for n from 0 to 8 do seq(T(n, k),k=0..n) end do; # yields sequence in triangular form
%Y Cf. A001003, A134425.
%K nonn,tabl
%O 0,2
%A _Emeric Deutsch_, Nov 05 2007
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