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A110438 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.

%I A110438 #28 May 04 2025 23:43:50
%S A110438 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 A110438 175,155,121,82,47,22,7,1,434,389,311,220,135,70,29,8,1,1082,979,799,
%U A110438 584,378,212,100,37,9,1,2709,2471,2051,1541,1039,620,320,138,46,10,1
%N A110438 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 A110438 The row sums are the even-indexed Fibonacci numbers.
%C A110438 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 A110438 A. Nkwanta, A Riordan matrix approach to unifying a selected class of combinatorial arrays, Congressus Numerantium, 160 (2003), pp. 33-55.
%D A110438 A. Nkwanta, A note on Riordan matrices, Contemporary Mathematics Series, AMS, 252 (1999), pp. 99-107.
%D A110438 A. Nkwanta, Lattice paths, generating functions and the Riordan group, Ph.D. Thesis, Howard University, Washington DC, 1997.
%H A110438 Andrew Howroyd, <a href="/A110438/b110438.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A110438 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.
%H A110438 Tian-Xiao He, <a href="https://doi.org/10.1016/j.disc.2019.111718">A-sequences, Z-sequence, and B-sequences of Riordan matrices</a>, Discrete Mathematics 343.3 (2020): 111718.
%F A110438 Recurrence is d(0, 0) = 1, d(1, 0) = 1, d(n+1, 0) = 2*d(n, 0) + Sum_{j>=1} d(n-j, j), n>=1 for leftmost column and d(n+1, k) = d(n, k-1) + d(n, k) + Sum_{j>=1} d(n-j, k+j), n>=2, k>=1 and n>j; Riordan array d(n, k): (((1-z)/(2*z))*(sqrt(1+z+z^2)/sqrt(1-3*z+z^2) - 1), ((1-z+z^2)-sqrt(1-2*z-z^2-2*z^3+z^4))/(2*z)).
%e A110438 Triangle starts:
%e A110438   1;
%e A110438   1,1;
%e A110438   2,2,1;
%e A110438   5,4,3,1;
%e A110438   12,10,7,4,1;
%p A110438 A110438 := proc (n, k)
%p A110438     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 A110438 end proc:
%p A110438 seq(seq(A110438(n, k), k = 0..n), n = 0..10); # _Peter Bala_, Jul 14 2021
%o A110438 (PARI) \\ ColGf gives g.f. of k-th column.
%o A110438 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 A110438 T(n,k) = {polcoef(ColGf(k,n), n)} \\ _Andrew Howroyd_, Mar 02 2023
%Y A110438 Row sums are A001519(n+1).
%Y A110438 Cf. A097724, A158793.
%K A110438 easy,nonn,tabl
%O A110438 0,4
%A A110438 Asamoah Nkwanta (Nkwanta(AT)jewel.morgan.edu), Aug 10 2005
%E A110438 Terms a(55) and beyond from _Andrew Howroyd_, Mar 02 2023