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%I #15 Aug 18 2017 03:14:12
%S 1,1,1,1,2,1,1,3,4,1,1,4,9,6,1,1,5,16,18,9,1,1,6,25,40,36,12,1,1,7,36,
%T 75,100,60,16,1,1,8,49,126,225,200,100,20,1,1,9,64,196,441,525,400,
%U 150,25,1,1,10,81,288,784,1176,1225,700,225,30,1,1,11,100,405,1296,2352
%N Triangle, read by rows, of the number of left factors of peakless Motzkin paths of length n having k number of U's and D's (i.e., number of paths from (0,0) to the line x=n, consisting of steps U=(1,1), H=(1,0), D=(1,1), that never go below the x-axis and a U step is never followed by a D step).
%C Row sums form A091964, the number of left factors of peakless Motzkin paths of length n.
%F G.f.: A(x, y) = 2/(1-x+x^2*y^2 - 2*x*y + sqrt((1-x+x^2*y^2)^2 - 4*x^2*y^2)) (due to _Emeric Deutsch_).
%F T(n, k) = C(n-floor(k/2), floor((k+1)/2))*C(n-floor((k+1)/2), floor(k/2)) = A104557(n, k)/(n-k)!.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 3, 4, 1;
%e 1, 4, 9, 6, 1;
%e 1, 5, 16, 18, 9, 1;
%e 1, 6, 25, 40, 36, 12, 1;
%e 1, 7, 36, 75, 100, 60, 16, 1;
%e 1, 8, 49, 126, 225, 200, 100, 20, 1; ...
%p T:=proc(n,k) if k<=n then binomial(n-floor(k/2),floor((k+1)/2))*binomial(n-floor((k+1)/2),floor(k/2)) else 0 fi end: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form # _Emeric Deutsch_, Mar 16 2005
%o (PARI) T(n,k)=binomial(n-(k\2),(k+1)\2)*binomial(n-((k+1)\2),k\2)
%o (PARI) {T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k));polcoeff(polcoeff( 2/(1-X+X^2*Y^2-2*X*Y+sqrt((1-X+X^2*Y^2)^2-4*X^2*Y^2)),n,x),k,y)}
%Y Cf. A091964, A104557.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_ and _Emeric Deutsch_, Mar 16 2005
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