A104559 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).

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, 75, 100, 60, 16, 1, 1, 8, 49, 126, 225, 200, 100, 20, 1, 1, 9, 64, 196, 441, 525, 400, 150, 25, 1, 1, 10, 81, 288, 784, 1176, 1225, 700, 225, 30, 1, 1, 11, 100, 405, 1296, 2352

Offset: 0

Author

Paul D. Hanna and Emeric Deutsch, Mar 16 2005

Keywords

• nonn
• tabl

Comments

Row sums form A091964, the number of left factors of peakless Motzkin paths of length n.

Examples

			Triangle begins:
  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,  75, 100,  60,  16,   1;
  1,   8,  49, 126, 225, 200, 100,  20,   1; ...
		

Crossrefs

Cf. A091964, A104557.

Programs

Maple

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

PARI

T(n,k)=binomial(n-(k\2),(k+1)\2)*binomial(n-((k+1)\2),k\2)

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)}

Formula

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).

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)!.