cp's OEIS Frontend

A145598 Triangular array of generalized Narayana numbers: T(n, k) = 4*binomial(n+1, k+3)*binomial(n+1, k-1)/(n+1).

%I A145598 #21 Aug 03 2025 16:09:19
%S A145598 1,4,4,10,24,10,20,84,84,20,35,224,392,224,35,56,504,1344,1344,504,56,
%T A145598 84,1008,3780,5760,3780,1008,84,120,1848,9240,19800,19800,9240,1848,
%U A145598 120,165,3168,20328,58080,81675,58080,20328,3168,165,220,5148,41184,151008
%N A145598 Triangular array of generalized Narayana numbers: T(n, k) = 4*binomial(n+1, k+3)*binomial(n+1, k-1)/(n+1).
%C A145598 T(n,k) is the number of walks of n unit steps, each step in the direction either up (U), down (D), right (R) or left (L), starting from (0,0) and finishing at lattice points on the horizontal line y = 3 and which remain in the upper half-plane y >= 0. An example is given in the Example section below.
%C A145598 The current array is the case r = 3 of the generalized Narayana numbers N_r(n,k) := (r + 1)/(n + 1)*binomial(n + 1,k + r)*binomial(n + 1,k - 1), which count walks of n steps from the origin to points on the horizontal line y = r that remain in the upper half-plane. Case r = 0 gives the table of Narayana numbers A001263 (but with an offset of 0 in the row numbering). For other cases see A145596 (r = 1), A145597 (r = 2) and A145599 (r = 4).
%H A145598 F. Cai, Q.-H. Hou, Y. Sun, and A. L. B. Yang, <a href="http://arxiv.org/abs/1808.05736">Combinatorial identities related to 2x2 submatrices of recursive matrices</a>, arXiv:1808.05736 [math.CO], 2018; Table 2.1 for k=3.
%H A145598 R. K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
%F A145598 T(n,k) = 4/(n+1)*binomial(n+1,k+3)*binomial(n+1,k-1) for n >=3 and 1 <= k <= n-2. In the notation of [Guy], T(n,k) equals w_n(x,y) at (x,y) = (2*k - n + 1,3). Row sums A003518.
%F A145598 O.g.f. for column k+2: 4/(k + 1) * y^(k+4)/(1 - y)^(k+6) * Jacobi_P(k,4,1,(1 + y)/(1 - y)).
%F A145598 Identities for row polynomials R_n(x) = Sum_{k = 1 .. n - 2} T(n,k)*x^k:
%F A145598 x^3*R_(n-1)(x) = 4*(n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)*(n + 4)) * Sum_{k = 0..n} binomial(n + 4,k) * binomial(2n - k,n) * (x - 1)^k;
%F A145598 Sum_{k = 1..n} (-1)^(k+1)*binomial(n,k)*R_k(x^2)*(1 + x)^(2*(n-k)) = R_n(1)*x^(n-1) = A003518(n)*x^(n-1).
%F A145598 Row generating polynomial R_(n+3)(x) = 4/(n+4)*x*(1-x)^n * Jacobi_P(n,4,4,(1+x)/(1-x)). - _Peter Bala_, Oct 31 2008
%F A145598 G.f.: A(x) = x*A145596(x)^2. - _Vladimir Kruchinin_, Oct 09 2020
%e A145598 Triangle starts
%e A145598   n\k|  1     2     3     4     5     6
%e A145598   =====================================
%e A145598    3 |  1
%e A145598    4 |  4     4
%e A145598    5 | 10    24    10
%e A145598    6 | 20    84    84    20
%e A145598    7 | 35   224   392   224    35
%e A145598    8 | 56   504  1344  1344   504    56
%e A145598   ...
%e A145598 Row 5: T(5,3) = 10: the 10 walks of length 5 from (0,0) to (2,3) are UUURR, UURUR, UURRU, URUUR, URURU, URRUU, RUUUR, RUURU, RURUU and RRUUU.
%e A145598 *
%e A145598 *......*......*......y......*......*......*
%e A145598 .
%e A145598 .
%e A145598 *.....10......*.....24......*.....10......*
%e A145598 .
%e A145598 .
%e A145598 *......*......*......*......*......*......*
%e A145598 .
%e A145598 .
%e A145598 *......*......*......*......*......*......*
%e A145598 .
%e A145598 .
%e A145598 *......*......*......o......*......*......* x axis
%e A145598 .
%p A145598 T := (n,k) -> 4/(n+1)*binomial(n+1,k+3)*binomial(n+1,k-1):
%p A145598 for n from 3 to 12 do seq(T(n, k), k = 1 .. n-2) end do;
%Y A145598 Cf. A003518 (row sums), A001263, A145602, A145596, A145597, A145599.
%K A145598 easy,nonn,tabl
%O A145598 3,2
%A A145598 _Peter Bala_, Oct 15 2008