A154825 -id:A154825 - cp's OEIS frontend

A086810 Triangle obtained by adding a leading diagonal 1,0,0,0,... to A033282.

1, 0, 1, 0, 1, 2, 0, 1, 5, 5, 0, 1, 9, 21, 14, 0, 1, 14, 56, 84, 42, 0, 1, 20, 120, 300, 330, 132, 0, 1, 27, 225, 825, 1485, 1287, 429, 0, 1, 35, 385, 1925, 5005, 7007, 5005, 1430, 0, 1, 44, 616, 4004, 14014, 28028, 32032, 19448, 4862, 0, 1, 54, 936, 7644, 34398, 91728

Offset: 0

Philippe Deléham, Aug 05 2003

Mirror image of triangle A133336. - Philippe Deléham, Dec 10 2008
From Tom Copeland, Oct 09 2011: (Start)
With polynomials
P(0,t) = 0
P(1,t) = 1
P(2,t) = t
P(3,t) = t + 2 t^2
P(4,t) = t + 5 t^2 +  5 t^3
P(5,t) = t + 9 t^2 + 21 t^3 + 14 t^4
The o.g.f. A(x,t) = {1+x-sqrt[(1-x)^2-4xt]}/[2(1+t)] (see Drake et al.).
B(x,t)= x-t x^2/(1-x)= x-t(x^2+x^3+x^4+...) is the comp. inverse in x.
Let h(x,t) = 1/(dB/dx) = (1-x)^2/(1+(1+t)*x*(x-2)) = 1/(1-t(2x+3x^2+4x^3+...)), an o.g.f. in x for row polynomials in t of A181289. Then P(n,t) is given by (1/n!)(h(x,t)*d/dx)^n x, evaluated at x=0, A = exp(x*h(y,t)*d/dy) y, eval. at y=0, and dA/dx = h(A(x,t),t). These results are a special case of A133437 with u(x,t) = B(x,t), i.e., u_1=1 and (u_n)=-t for n > 1. See A001003 for t = 1. (End)
Let U(x,t) = [A(x,t)-x]/t, then U(x,0) = -dB(x,t)/dt and U satisfies dU/dt = UdU/dx, the inviscid Burgers' equation (Wikipedia), also called the Hopf equation (see Buchstaber et al.). Also U(x,t) = U(A(x,t),0) = U(x+tU,0) since U(x,0) = [x-B(x,t)]/t. - Tom Copeland, Mar 12 2012
Diagonals of A132081 are essentially rows of this sequence. - Tom Copeland, May 08 2012
T(r, s) is the number of [0,r]-covering hierarchies with s segments (see Kreweras). - Michel Marcus, Nov 22 2014
From Yu Hin Au, Dec 07 2019: (Start)
T(n,k) is the number of small Schröder n-paths (lattice paths from (0,0) to (2n,0) using steps U=(1,1), F=(2,0), D=(1,-1) with no F step on the x-axis) that has exactly k U steps.
T(n,k) is the number of Schröder trees (plane rooted tree where each internal node has at least two children) with exactly n+1 leaves and k internal nodes. (End)

			Triangle starts:
  1;
  0,  1;
  0,  1,  2;
  0,  1,  5,  5;
  0,  1,  9, 21, 14;
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
Yu Hin (Gary) Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv:1912.00555 [math.CO], 2019.
Paul Barry, On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.6.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
V. Buchstaber and E. Bunkova,Elliptic formal group laws, integral Hirzebruch genera and Kirchever genera,, arXiv:1010.0944 [math-ph], 2010 (see p. 19).
V. Buchstaber and T. Panov, Toric Topology. Chapter 1: Geometry and Combinatorics of Polytopes,, arXiv:1102.1079 [math.CO], 2011-2012 (see p. 41).
G. Chatel, V. Pilaud, Cambrian Hopf Algebras, arXiv:1411.3704 [math.CO], 2014-2015.
T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015.
T. Copeland, Compositional inverse pairs, the Burgers-Hopf equation, and the Stasheff associahedra, 2014.
T. Copeland, Lagrange a la Lah, 2011.
B. Drake, Ira M. Gessel, and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), Article 07.3.7.
G. Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973, p. 21-22.
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
J. Zhou, Quantum deformation theory of the Airy curve and the mirror symmetry of a point, arXiv preprint arXiv:1405.5296 [math.AG], 2014.

Diagonals: A000007, A000012, A000096, A033275, A033276, A033277, A033278, A033279, A000108, A002054, A002055, A002056, A007160, A033280, A033281.
The diagonals (except for A000007) are also the diagonals of A033282.
Row sums: A001003 (Schroeder numbers).
Cf. A033282, A084938.
Cf. A001003, A008297, A021009, A132081, A133437, A181289.

Table[Boole[n == 2] + If[# == -1, 0, Binomial[n - 3, #] Binomial[n + # - 1, #]/(# + 1)] &[k - 1], {n, 2, 12}, {k, 0, n - 2}] // Flatten (* after Jean-François Alcover at A033282, or *)
Table[If[n == 0, 1, Binomial[n, k] Binomial[n + k, k - 1]/n], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 22 2016 *)

t(n, k) = if (n==0, 1, binomial(n, k)*binomial(n+k, k-1)/n);
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n,k), ", ");); print(););} \\ Michel Marcus, Nov 22 2014

Triangle T(n, k) read by rows; given by [0, 1, 0, 1, 0, 1, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is Deléham's operator defined in A084938.
For k>0, T(n, k) = binomial(n-1, k-1)*binomial(n+k, k)/(n+1); T(0, 0) = 1 and T(n, 0) = 0 if n > 0. [corrected by Marko Riedel, May 04 2023]
Sum_{k>=0} T(n, k)*2^k = A107841(n). - Philippe Deléham, May 26 2005
Sum_{k>=0} T(n-k, k) = A005043(n). - Philippe Deléham, May 30 2005
T(n, k) = A108263(n+k, k). - Philippe Deléham, May 30 2005
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A001003(n), A107841(n), A131763(n), A131765(n), A131846(n), A131926(n), A131869(n), A131927(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. - Philippe Deléham, Nov 05 2007
Sum_{k=0..n} T(n,k)*5^k*(-2)^(n-k) = A152601(n). - Philippe Deléham, Dec 10 2008
Sum_{k=0..n} T(n,k)*(-1)^k*3^(n-k) = A154825(n). - Philippe Deléham, Jan 17 2009
Umbrally, P(n,t) = Lah[n-1,-t*a.]/n! = (1/n)*Sum_{k=1..n-1} binomial(n-2,k-1)a_k t^k/k!, where (a.)^k = a_k = (n-1+k)!/(n-1)!, the rising factorial, and Lah(n,t) = n!*Laguerre(n,-1,t) are the Lah polynomials A008297 related to the Laguerre polynomials of order -1. - Tom Copeland, Oct 04 2014
T(n, k) = binomial(n, k)*binomial(n+k, k-1)/n, for k >= 0; T(0, 0) = 1 (see Kreweras, p. 21). - Michel Marcus, Nov 22 2014
P(n,t) = Lah[n-1,-:Dt:]/n! t^(n-1) with (:Dt:)^k = (d/dt)^k t^k = k! Laguerre(k,0,-:tD:) with (:tD:)^j = t^j D^j. The normalized Laguerre polynomials of 0 order are given in A021009. - Tom Copeland, Aug 22 2016

Typo in a(60) corrected by Michael De Vlieger, Nov 21 2019

A091593 Reversion of Jacobsthal numbers A001045.

Original entry on oeis.org

1, -1, -1, 5, -3, -21, 51, 41, -391, 407, 1927, -6227, -2507, 49347, -71109, -236079, 966129, 9519, -7408497, 13685205, 32079981, -167077221, 60639939, 1209248505, -2761755543, -4457338681, 30629783831, -22124857219, -206064020315, 572040039283, 590258340811

Offset: 0

Paul Barry, Jan 23 2004

Hankel transform is (-2)^C(n+1,2). - Paul Barry, Apr 28 2009

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for reversions of series

Cf. A001263, A154825.

a := n -> hypergeom([-n,-n-1], [2], -2);
seq(simplify(a(n)), n=0..30); # Peter Luschny, Sep 22 2014
# Using function CompInv from A357588.
CompInv(25, n -> (2^n - (-1)^n)/3 ); # Peter Luschny, Oct 07 2022

a[n_] := Hypergeometric2F1[-n - 1, -n - 1, 2, -2] + (n + 1)*Hypergeometric2F1[-n, -n, 3, -2]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 03 2016, after Vladimir Kruchinin *)

a(n) := hypergeometric([ -n - 1, -n - 1 ], [ 2 ], -2) + (n + 1) * hypergeometric([ -n, -n ], [ 3 ], -2); /* Vladimir Kruchinin, Oct 12 2011 */

# Algorithm of L. Seidel (1877)
def A091593_list(n) :
    D = [0]*(n+2); D[1] = 1
    R = []; b = false; h = 1
    for i in range(2*n) :
        if b :
            for k in range(1, h, 1) : D[k] += -2*D[k+1]
            R.append(D[1])
        else :
            for k in range(h, 0, -1) : D[k] += D[k-1]
            h += 1
        b = not b
    return R
A091593_list(30)  # Peter Luschny, Oct 19 2012

G.f.: (-(1+x)+sqrt(1+2*x+9*x^2))/(4*x^2). - Corrected by Seiichi Manyama, May 12 2019
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*C(k)*(-1)^(n-k)*2^k, where C(n) is A000108(n). - Paul Barry, May 16 2005
G.f.: 1/(1+x+2x^2/(1+x+2x^2/(1+x+2x^2/(1+x+2x^2/(1+ ... (continued fraction). - Paul Barry, Apr 28 2009
a(n) = Sum_{i=0..n} (2^(i)*(-1)^(n-i)*binomial(n+1,i)^2*(n-i+1)/(i+1))/(n+1). - Vladimir Kruchinin, Oct 12 2011
Conjecture: (n+2)*a(n) +(2*n+1)*a(n-1) +9*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 26 2012
a(n) = (-1)^n*hypergeom([-n/2, (1-n)/2], [2], -8). - Peter Luschny, May 28 2014
R. J. Mathar's conjecture confirmed by Maple using this hypergeom form. - Robert Israel, Sep 22 2014
a(n) = Sum_{k = 0..n} (-2)^k * (1/(n+1))*binomial(n+1, k)*binomial(n+1, k+1) =  Sum_{k = 0..n} (-2)^k * N(n+1,k+1), where N(n,k) = A001263(n,k) are the Narayana numbers. - Peter Bala, Sep 01 2023

A336709 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} (-2)^(n-j) * binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 1, -2, 1, 1, -1, 2, 1, 1, 0, -1, 4, 1, 1, 1, -1, 5, -24, 1, 1, 2, 2, 0, -3, 48, 1, 1, 3, 8, 5, 2, -21, 24, 1, 1, 4, 17, 36, 13, 0, 51, -464, 1, 1, 5, 29, 109, 177, 36, -5, 41, 1376, 1, 1, 6, 44, 240, 766, 922, 104, 0, -391, -704

Offset: 0

Seiichi Manyama, Aug 01 2020

			Square array begins:
    1,   1,  1,  1,   1,    1,     1, ...
    1,   1,  1,  1,   1,    1,     1, ...
   -2,  -1,  0,  1,   2,    3,     4, ...
    2,  -1, -1,  2,   8,   17,    29, ...
    4,   5,  0,  5,  36,  109,   240, ...
  -24,  -3,  2, 13, 177,  766,  2177, ...
   48, -21,  0, 36, 922, 5699, 20910, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-3 give: A307969(n-1), (-1)^n * A154825(n), A090192, A246555.
Main diagonal gives A336714.
Cf. A336575, A336706, A336707, A336708.

T[0, k_] := 1; T[n_, k_] := Sum[(-2)^(n - j) * Binomial[n, j] * Binomial[n + (k - 1)*j, j - 1], {j, 1, n}] / n; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 01 2020 *)

{T(n, k) = if(n==0, 1, sum(j=1, n, (-2)^(n-j)*binomial(n, j)*binomial(n+(k-1)*j, j-1))/n)}

{T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k/(1+2*x*A)); polcoef(A, n)}

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k / (1 + 2 * x * A_k(x)).

A336727 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} (-k)^(n-j) * binomial(n,j) * binomial(n,j-1) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -1, 1, 1, 1, -2, -1, 0, 1, 1, 1, -3, 1, 5, 2, 1, 1, 1, -4, 5, 10, -3, 0, 1, 1, 1, -5, 11, 9, -38, -21, -5, 1, 1, 1, -6, 19, -4, -103, 28, 51, 0, 1, 1, 1, -7, 29, -35, -174, 357, 289, 41, 14, 1, 1, 1, -8, 41, -90, -203, 1176, -131, -1262, -391, 0, 1

Offset: 0

Seiichi Manyama, Aug 02 2020

			  1,  1,   1,   1,    1,    1,    1, ...
  1,  1,   1,   1,    1,    1,    1, ...
  1,  0,  -1,  -2,   -3,   -4,   -5, ...
  1, -1,  -1,   1,    5,   11,   19, ...
  1,  0,   5,  10,    9,   -4,  -35, ...
  1,  2,  -3, -38, -103, -174, -203, ...
  1,  0, -21,  28,  357, 1176, 2575, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-3 give: A000012, A090192, (-1)^n * A154825(n), A336729.
Main diagonal gives A336728.
Cf. A243631, A307884, A336708, A336709.

T[0, k_] := 1; T[n_, k_] := Sum[If[k == 0, Boole[n == j],(-k)^(n - j)] * Binomial[n, j] * Binomial[n , j - 1], {j, 1, n}] / n; Table[T[k, n- k], {n, 0, 11}, {k, 0, n}] //Flatten (* Amiram Eldar, Aug 02 2020 *)

{T(n, k) = if(n==0, 1, sum(j=1, n, (-k)^(n-j)*binomial(n, j)*binomial(n, j-1))/n)}

{T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A/(1+k*x*A)); polcoef(A, n)}

{T(n, k) = sum(j=0, n, (-k)^j*(k+1)^(n-j)*binomial(n, j)*binomial(n+j, n)/(j+1))}

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x) / (1 + k * x * A_k(x)).
A_k(x) = 2/(1 - (k+1)*x + sqrt(1 + 2*(k-1)*x + ((k+1)*x)^2)).
T(n, k) = Sum_{j=0..n} (-k)^j * (k+1)^(n-j) * binomial(n,j) * binomial(n+j,n)/(j+1).
(n+1) * T(n,k) = -(k-1) * (2*n-1) * T(n-1,k) - (k+1)^2 * (n-2) * T(n-2,k) for n>1. - Seiichi Manyama, Aug 08 2020

A352687 Triangle read by rows, a Narayana related triangle whose rows are refinements of twice the Catalan numbers (for n >= 2).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 4, 1, 0, 1, 7, 12, 7, 1, 0, 1, 11, 30, 30, 11, 1, 0, 1, 16, 65, 100, 65, 16, 1, 0, 1, 22, 126, 280, 280, 126, 22, 1, 0, 1, 29, 224, 686, 980, 686, 224, 29, 1, 0, 1, 37, 372, 1512, 2940, 2940, 1512, 372, 37, 1

Offset: 0

Peter Luschny, Apr 26 2022

This is the second triangle in a sequence of Narayana triangles. The first is A090181, whose n-th row is a refinement of Catalan(n), whereas here the n-th row of T is a refinement of 2*Catalan(n-1). We can show that T(n, k) <= A090181(n, k) for all n, k. The third triangle in this sequence is A353279, where also a recurrence for the general case is given.
Here we give a recurrence for the row polynomials, which correspond to the recurrence of the classical Narayana polynomials combinatorially proved by Sulanke (see link).
The polynomials have only real zeros and form a Sturm sequence. This follows from the recurrence along the lines given in the Chen et al. paper.
Some interesting sequences turn out to be the evaluation of the polynomial sequence at a fixed point (see the cross-references), for example the reversion of the Jacobsthal numbers A001045 essentially is -(-2)^n*P(n, -1/2).
The polynomials can also be represented as the difference between generalized Narayana polynomials, see the formula section.

			Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,  1;
[3] 0, 1,  2,   1;
[4] 0, 1,  4,   4,   1;
[5] 0, 1,  7,  12,   7,   1;
[6] 0, 1, 11,  30,  30,  11,   1;
[7] 0, 1, 16,  65, 100,  65,  16,   1;
[8] 0, 1, 22, 126, 280, 280, 126,  22,  1;
[9] 0, 1, 29, 224, 686, 980, 686, 224, 29, 1;

Xi Chen, Arthur Li Bo Yang, James Jing Yu Zhao, Recurrences for Callan's Generalization of Narayana Polynomials, J. Syst. Sci. Complex (2021).
Peter Luschny, Illustration of the polynomials.
Robert A. Sulanke, The Narayana distribution, J. Statist. Plann. Inference, 2002, 101: 311-326, formula 2.

Cf. A090181 and A001263 (Narayana), A353279 (case 3), A000108 (Catalan), A145596, A172392 (central terms), A000124 (subdiagonal, column 2), A115143.
Essentially twice the Catalan numbers: A284016 (also A068875, A002420).
Values of the polynomial sequence: A068875 (row sums): P(1), A154955: P(-1), A238113: P(2)/2, A125695 (also A152681): P(-2), A054872: P(3)/2, P(3)/6 probable A234939, A336729: P(-3)/6, A082298: P(4)/5, A238113: 2^n*P(1/2), A154825 and A091593: 2^n*P(-1/2).

T := (n, k) -> if n = k then 1 elif k = 0 then 0 else
binomial(n, k)^2*(k*(2*k^2 + (n + 1)*(n - 2*k))) / (n^2*(n - 1)*(n - k + 1)) fi:
seq(seq(T(n, k), k = 0..n), n = 0..10);
# Alternative:
gf := 1 - x + (1 + y)*(1 - x*(y - 1) - sqrt((x*y + x - 1)^2 - 4*x^2*y))/2:
serx := expand(series(gf, x, 16)): coeffy := n -> coeff(serx, x, n):
seq(seq(coeff(coeffy(n), y, k), k = 0..n), n = 0..10);
# Using polynomial recurrence:
P := proc(n, x) option remember; if n < 3 then [1, x, x + x^2] [n + 1] else
((2*n - 3)*(x + 1)*P(n - 1, x) - (n - 3)*(x - 1)^2*P(n - 2, x)) / n fi end:
Trow := n -> seq(coeff(P(n, x), x, k), k = 0..n): seq(Trow(n), n = 0..10);
# Represented by generalized Narayana polynomials:
N := (n, k, x) -> add(((k+1)/(n-k))*binomial(n-k,j-1)*binomial(n-k,j+k)*x^(j+k), j=0..n-2*k): seq(print(ifelse(n=0, 1, expand(N(n,0,x) - N(n,1,x)))), n=0..7);

H[0, ] := 1; H[1, x] := x;
H[n_, x_] := x*(x + 1)*Hypergeometric2F1[1 - n, 2 - n, 2, x];
Hrow[n_] := CoefficientList[H[n, x], x]; Table[Hrow[n], {n, 0, 9}] // TableForm

from math import comb as binomial
def T(n, k):
    if k == n: return 1
    if k == 0: return 0
    return ((binomial(n, k)**2 * (k * (2 * k**2 + (n + 1) * (n - 2 * k))))
           // (n**2 * (n - 1) * (n - k + 1)))
def Trow(n): return [T(n, k) for k in range(n + 1)]
for n in range(10): print(Trow(n))

# The recursion with cache is (much) faster:
from functools import cache
@cache
def T_row(n):
    if n < 3: return ([1], [0, 1], [0, 1, 1])[n]
    A = T_row(n - 2) + [0, 0]
    B = T_row(n - 1) + [1]
    for k in range(n - 1, 1, -1):
        B[k] = (((B[k] + B[k - 1]) * (2 * n - 3)
               - (A[k] - 2 * A[k - 1] + A[k - 2]) * (n - 3)) // n)
    return B
for n in range(10): print(T_row(n))

Explicit formula (additive form):
T(n, n) = 1, T(n > 0, 0) = 0 and otherwise T(n, k) = binomial(n, k)*binomial(n - 1, k - 1)/(n - k + 1) - 2*binomial(n - 1, k)*binomial(n - 1, k - 2)/(n - 1).
Multiplicative formula with the same boundary conditions:
T(n, k) = binomial(n, k)^2*(k*(2*k^2 + (n + 1)*(n - 2*k)))/(n^2*(n-1)*(n- k + 1)).
Bivariate generating function:
T(n, k) = [x^n] [y^k](1 - x + (1+y)*(1-x*(y-1) - sqrt((x*y+x-1)^2 - 4*x^2*y))/2).
Recursion based on polynomials:
T(n, k) = [x^k] (((2*n - 3)*(x + 1)*P(n - 1, x) - (n - 3)*(x - 1)^2*P(n - 2, x)) / n) with P(0, x) = 1, P(1, x) = x, and P(2, x) = x + x^2.
Recursion based on rows (see the second Python program):
T(n, k) = (((B(k) + B(k-1)) * (2*n - 3) - (A(k) - 2*A(k-1) + A(k-2))*(n-3))/n), where A(k) = T(n-2, k) and B(k) = T(n-1, k), for n >= 3.
Hypergeometric representation:
T(n, k) = [x^k] x*(x + 1)*hypergeom([1 - n, 2 - n], [2], x) for n >= 2.
Row sums:
Sum_{k=0..n} T(n, k) = (2/n)*binomial(2*(n - 1), n - 1) = A068875(n-1) for n >= 2.
A generalization of the Narayana polynomials is given by
N{n, k}(x) = Sum_{j=0..n-2*k}(((k + 1)/(n - k)) * binomial(n - k, j - 1) * binomial(n - k, j + k) * x^(j + k)).
N{n, 0}(x) are the classical Narayana polynomials A001263 and N{n, 1}(x) is a shifted version of A145596 based in (3, 2). Our polynomials are the difference P(n, x) = N{n, 0}(x) - N{n, 1}(x) for n >= 1.
Let RS(T, n) denote the row sum of the n-th row of T, then RS(T, n) - RS(A090181, n) = -4*binomial(2*n - 3, n - 3)/(n + 1) = A115143(n + 1) for n >= 3.

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A086810 Triangle obtained by adding a leading diagonal 1,0,0,0,... to A033282.

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A091593 Reversion of Jacobsthal numbers A001045.

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A336709 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} (-2)^(n-j) * binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.

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A336727 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} (-k)^(n-j) * binomial(n,j) * binomial(n,j-1) for n > 0.

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A352687 Triangle read by rows, a Narayana related triangle whose rows are refinements of twice the Catalan numbers (for n >= 2).

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