A152659 Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) and having k turns (NE or EN) (1<=k<=2n-1).

2, 2, 2, 2, 2, 4, 8, 4, 2, 2, 6, 18, 18, 18, 6, 2, 2, 8, 32, 48, 72, 48, 32, 8, 2, 2, 10, 50, 100, 200, 200, 200, 100, 50, 10, 2, 2, 12, 72, 180, 450, 600, 800, 600, 450, 180, 72, 12, 2, 2, 14, 98, 294, 882, 1470, 2450, 2450, 2450, 1470, 882, 294, 98, 14, 2, 2, 16, 128, 448

Offset: 1

Author

Emeric Deutsch, Dec 10 2008

Keywords

• nonn
• tabf

Comments

Row n has 2n-1 entries.

Sum of entries of row n = binomial(2n,n) = A000984(n) (the central binomial coefficients).

Sum(k*T(n,k),k=0..2n-1) = n*binomial(2n,n) = A005430(n).

Examples

			T(3,2)=4 because we have ENNNEE, EENNNE, NEEENN and NNEEEN.
Triangle starts:
  2;
  2,2,2;
  2,4,8,4,2;
  2,6,18,18,18,6,2;
  2,8,32,48,72,48,32,8,2;
  ...
		

Links

• Kent E. Morrison, Matching adjacent cards, arXiv:2501.07753 [math.CO], 2025. See alpha_r p. 15.

Crossrefs

Cf. A000984, A005430.

Programs

Maple

T := proc (n, k) if `mod`(k, 2) = 0 then 2*binomial(n-1, (1/2)*k-1)*binomial(n-1, (1/2)*k) else 2*binomial(n-1, (1/2)*k-1/2)^2 end if end proc: for n to 9 do seq(T(n, k), k = 1 .. 2*n-1) end do; # yields sequence in triangular form

Formula

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

T(n,2k-1) = 2*binomial(n-1,k-1)^2.

G.f.: [1+t*r(t^2,z)]/[1-t*r(t^2,z)], where r(t,z) is the Narayana function, defined by r = z(1+r)(1+tr).