%I #20 Mar 02 2023 16:53:28
%S 1,1,1,2,2,1,5,4,3,1,12,10,7,4,1,29,25,18,11,5,1,71,62,47,30,16,6,1,
%T 175,155,121,82,47,22,7,1,434,389,311,220,135,70,29,8,1,1082,979,799,
%U 584,378,212,100,37,9,1,2709,2471,2051,1541,1039,620,320,138,46,10,1
%N Triangular array giving the number of NSEW unit step lattice paths of length n with terminal height k subject to the following restrictions. The paths start at the origin (0,0) and take unit steps (0,1)=N(north), (0,-1)=S(south), (1,0)=E(east) and (-1,0)=W(west) such that no paths pass below the x-axis, no paths begin with W, all W steps remain on the x-axis and there are no NS steps.
%C The row sums are the even-indexed Fibonacci numbers.
%C Matrix product Q^(-1) * P * Q, where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158793. - _Peter Bala_, Jul 14 2021
%D He, Tian-Xiao. "A-sequences, Z-sequence, and B-sequences of Riordan matrices." Discrete Mathematics 343.3 (2020): 111718.
%D A. Nkwanta, A Riordan matrix approach to unifying a selected class of combinatorial arrays, Congressus Numerantium, 160 (2003), pp. 33-55.
%D A. Nkwanta, A note on Riordan matrices, Contemporary Mathematics Series, AMS, 252 (1999), pp. 99-107.
%D A. Nkwanta, Lattice paths, generating functions and the Riordan group, Ph.D. Thesis, Howard University, Washington DC, 1997.
%H Andrew Howroyd, <a href="/A110438/b110438.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H Naiomi T. Cameron and Asamoah Nkwanta, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Cameron/cameron46.html">On Some (Pseudo) Involutions in the Riordan Group</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
%F Recurrence is d(0, 0)= 1, d(1, 0)=1, d(n+1, 0) = 2*d(n, 0) + sum(d(n-j, j)j>=1, n>=1 for leftmost column and d(n+1, k) = d(n, k-1) + d(n, k) + sum(d(n-j, k+j)j>=1, n>=2, k>=1 and n>j; Riordan array d(n, k): (((1-z)/2z)*(sqrt(1+z+z^2)/sqrt(1-3z+z^2) -1), ((1-z+z^2)-sqrt(1-2z-z^2-2z^3+z^4)/2z)).
%e Triangle starts:
%e 1;
%e 1,1;
%e 2,2,1;
%e 5,4,3,1;
%e 12,10,7,4,1;
%p A110438 := proc (n, k)
%p add((-1)^binomial(n-i+1, 2)*binomial(floor((1/2)*n+(1/2)*i), i)*add(binomial(i, j)*binomial(j, floor((1/2)*j-(1/2)*k)), j = k..i), i = 0..n);
%p end proc:
%p seq(seq(A110438(n, k), k = 0..n), n = 0..10); # _Peter Bala_, Jul 14 2021
%o (PARI) \\ ColGf gives g.f. of k-th column.
%o ColGf(k,n)={my(g=(1 - x + x^2 - sqrt(1 - 2*x - x^2 - 2*x^3 + x^4 + O(x^(n-k+3))))/(2*x^2)); (1 - x)*g/(1 - x*g)*(x*g)^k}
%o T(n,k) = {polcoef(ColGf(k,n), n)} \\ _Andrew Howroyd_, Mar 02 2023
%Y Row sums are A001519(n+1).
%Y Cf. A097724, A158793.
%K easy,nonn,tabl
%O 0,4
%A Asamoah Nkwanta (Nkwanta(AT)jewel.morgan.edu), Aug 10 2005
%E Terms a(55) and beyond from _Andrew Howroyd_, Mar 02 2023