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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).

%I A152659 #9 Jan 15 2025 01:44:38
%S A152659 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,
%T A152659 100,200,200,200,100,50,10,2,2,12,72,180,450,600,800,600,450,180,72,
%U A152659 12,2,2,14,98,294,882,1470,2450,2450,2450,1470,882,294,98,14,2,2,16,128,448
%N 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).
%C A152659 Row n has 2n-1 entries.
%C A152659 Sum of entries of row n = binomial(2n,n) = A000984(n) (the central binomial coefficients).
%C A152659 Sum(k*T(n,k),k=0..2n-1) = n*binomial(2n,n) = A005430(n).
%H A152659 Kent E. Morrison, <a href="https://arxiv.org/abs/2501.07753">Matching adjacent cards</a>, arXiv:2501.07753 [math.CO], 2025. See alpha_r p. 15.
%F A152659 T(n,2k) = 2*binomial(n-1,k-1)*binomial(n-1,k);
%F A152659 T(n,2k-1) = 2*binomial(n-1,k-1)^2.
%F A152659 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).
%e A152659 T(3,2)=4 because we have ENNNEE, EENNNE, NEEENN and NNEEEN.
%e A152659 Triangle starts:
%e A152659   2;
%e A152659   2,2,2;
%e A152659   2,4,8,4,2;
%e A152659   2,6,18,18,18,6,2;
%e A152659   2,8,32,48,72,48,32,8,2;
%e A152659   ...
%p A152659 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
%Y A152659 Cf. A000984, A005430.
%K A152659 nonn,tabf
%O A152659 1,1
%A A152659 _Emeric Deutsch_, Dec 10 2008