A080245 -id:A080245 - cp's OEIS frontend

A080247 Formal inverse of triangle A080246. Unsigned version of A080245.

1, 2, 1, 6, 4, 1, 22, 16, 6, 1, 90, 68, 30, 8, 1, 394, 304, 146, 48, 10, 1, 1806, 1412, 714, 264, 70, 12, 1, 8558, 6752, 3534, 1408, 430, 96, 14, 1, 41586, 33028, 17718, 7432, 2490, 652, 126, 16, 1, 206098

Offset: 0

Paul Barry, Feb 15 2003

Row sums are little Schroeder numbers A001003. Diagonal sums are generalized Fibonacci numbers A006603. Columns include A006318, A006319, A006320, A006321.
T(n,k) is the number of dissections of a convex (n+3)-gon by nonintersecting diagonals with exactly k diagonals emanating from a fixed vertex. Example: T(2,1)=4 because the dissections of the convex pentagon ABCDE having exactly one diagonal emanating from the vertex A are: {AC}, {AD}, {AC,EC} and {AD,BD}. - Emeric Deutsch, May 31 2004
For more triangle sums, see A180662, see the Schroeder triangle A033877 which is the mirror of this triangle. - Johannes W. Meijer, Jul 15 2013

			Triangle starts:
[0]    1
[1]    2,    1
[2]    6,    4,   1
[3]   22,   16,   6,   1
[4]   90,   68,  30,   8,  1
[5]  394,  304, 146,  48, 10,  1
[6] 1806, 1412, 714, 264, 70, 12, 1
...
From _Gary W. Adamson_, Jul 25 2011: (Start)
n-th row = top row of M^n, M = the following infinite square production matrix:
  2, 1, 0, 0, 0, ...
  2, 2, 1, 0, 0, ...
  2, 2, 2, 1, 0, ...
  2, 2, 2, 2, 1, ...
  ... (End)

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Paul Barry, Laurent Biorthogonal Polynomials and Riordan Arrays, arXiv preprint arXiv:1311.2292 [math.CA], 2013.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 21, 29.
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.
Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.
W.-j. Woan, The Lagrange inversion formula and divisibility properties, JIS 10 (2007) 07.7.8, example 5.

Cf. A000007, A033877 (mirror), A084938.

A080247:=(n,k)->(k+1)*add(binomial(n+1,k+j+1)*binomial(n+j,j),j=0..n-k)/(n+1):
seq(seq(A080247(n,k),k=0..n),n=0..9);

Clear[w] w[n_, k_] /; k < 0 || k > n := 0 w[0,0]=1 ; w[n_, k_] /; 0 <= k <= n && !n == k == 0 := w[n, k] = w[n-1,k-1] + w[n-1,k] + w[n,k+1] Table[w[n,k],{n,0,10},{k,0,n}] (* David Callan, Jul 03 2006 *)
T[n_, k_] := Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, -1];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Peter Luschny, Jan 08 2018 *)

T(n,k):=((k+1)*sum(2^m*binomial(n+1,m)*binomial(n-k-1,n-k-m),m,0,n-k))/(n+1); /* Vladimir Kruchinin, Jan 10 2022 */

def A080247_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)-2*sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^(n-k)*prec(n, k) for k in (1..n)]
for n in (1..10): print(A080247_row(n)) # Peter Luschny, Mar 16 2016

G.f.: 2/(2+y*x-y+y*(x^2-6*x+1)^(1/2))/y/x. - Vladeta Jovovic, Feb 16 2003
Essentially same triangle as triangle T(n,k), n > 0 and k > 0, read by rows; given by [0, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] DELTA A000007 where DELTA is Deléham's operator defined in A084938.
T(n, k) = T(n-1, k-1) + 2*Sum_{j>=0} T(n-1, k+j) with T(0, 0) = 1 and T(n, k)=0 if k < 0. - Philippe Deléham, Jan 19 2004
T(n, k) = (k+1)*Sum_{j=0..n-k} (binomial(n+1, k+j+1)*binomial(n+j, j))/(n+1). - Emeric Deutsch, May 31 2004
Recurrence: T(0,0)=1; T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n,k+1). - David Callan, Jul 03 2006
T(n, k) = binomial(n, k)*hypergeom([k - n, n + 1], [k + 2], -1). - Peter Luschny, Jan 08 2018
T(n,k) = (k+1)/(n+1)*Sum_{m=0..n-k} 2^m*binomial(n+1,m)*binomial(n-k-1,n-k-m). - Vladimir Kruchinin, Jan 10 2022
From Peter Bala, Sep 16 2024: (Start)
Riordan array (S(x), x*S(x)), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the g.f. of the large Schröder numbers A006318.
For integer m and n >= 1, (m + 2)*[x^n] S(x)^(m*n) = m*[x^n] (1/S(-x))^((m+2)*n). For cases of this identity see A103885 (m = 1), A333481 (m = 2) and A370102 (m = 3). (End)

A033877 Triangular array read by rows associated with Schroeder numbers: T(1,k) = 1; T(n,k) = 0 if k < n; T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k).

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 6, 16, 22, 1, 8, 30, 68, 90, 1, 10, 48, 146, 304, 394, 1, 12, 70, 264, 714, 1412, 1806, 1, 14, 96, 430, 1408, 3534, 6752, 8558, 1, 16, 126, 652, 2490, 7432, 17718, 33028, 41586, 1, 18, 160, 938, 4080, 14002, 39152, 89898, 164512, 206098

Offset: 1

N. J. A. Sloane

A106579 is in some ways a better version of this sequence, but since this was entered first it will be the main entry for this triangle.
The diagonals of this triangle are self-convolutions of the main diagonal A006318: 1, 2, 6, 22, 90, 394, 1806, ... . - Philippe Deléham, May 15 2005
From Johannes W. Meijer, Sep 22 2010, Jul 15 2013: (Start)
Note that for the terms T(n,k) of this triangle n indicates the column and k the row.
The triangle sums, see A180662, link Schroeder's triangle with several sequences, see the crossrefs. The mirror of this triangle is A080247.
Quite surprisingly the Kn1p sums, p >= 1, are all related to A026003 and crystal ball sequences for n-dimensional cubic lattices (triangle offset is 0): Kn11(n) = A026003(n), Kn12(n) = A026003(n+2) - 1, Kn13(n) = A026003(n+4) - A005408(n+3), Kn14(n) =  A026003(n+6) - A001844(n+4), Kn15(n) = A026003(n+8) - A001845(n+5), Kn16(n) = A026003(n+10) - A001846(n+6), Kn17(n) = A026003(n+12) - A001847(n+7), Kn18(n) = A026003(n+14) - A001848(n+8), Kn19(n) = A026003(n+16) - A001849(n+9), Kn110(n) = A026003(n+18) - A008417(n+10), Kn111(n) = A026003(n+20) - A008419(n+11), Kn112(n) = A026003(n+22) - A008421(n+12). (End)
T(n,k) is the number of normal semistandard Young tableaux with two columns, one of height k and one of height n. The recursion can be seen by performing jeu de taquin deletion on all instances of the smallest value. (If there are two instances of the smallest value, jeu de taquin deletion will always shorten the right column first and the left column second.) - Jacob Post, Jun 19 2018

			Triangle starts:
  1;
  1,    2;
  1,    4,    6;
  1,    6,   16,   22;
  1,    8,   30,   68,   90;
  1,   10,   48,  146,  304,  394;
  1,   12,   70,  264,  714, 1412, 1806;
  ... - _Joerg Arndt_, Sep 29 2013

T. D. Noe, Rows k = 1..50 of triangle, flattened
Henry Bottomley, Illustration of initial terms
Kevin Brown, Hipparchus on Compound Statements, 1994-2010. - _Johannes W. Meijer_, Sep 22 2010
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
G. Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.
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)
G. Kreweras, Aires des chemins surdiagonaux et application à un problème économique, Cahiers du Bureau universitaire de recherche opérationnelle Série Recherche 24 (1976): 1-8. [Annotated scanned copy]
J. W. Meijer, Famous numbers on a chessboard, Acta Nova, Volume 4, No.4, December 2010. pp. 589-598.
J. M. Oh, An explicit formula for the number of fuzzy subgroups of a finite abelian p-group of rank two, Iranian Journal of Fuzzy Systems, Dec 2013, Vol. 10 Issue 6, pp. 125-135.
E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.
S. Samieinia, The number of continuous curves in digital geometry, Port. Math. 67 (1) (2010) 75-89, last table.
R. A. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (2003), Article 03.1.5, 19 pp.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

Essentially same triangle as A080247 and A080245 but with rows read in reversed order. Also essentially the same triangle as A106579.
Cf. A000007, A006318, A006319, A006320, A006321, A008288, A080243.
Cf. A084938, A103885, A238112, A330801.
Cf. A001003 (row sums), A026003 (antidiagonal sums).
Triangle sums (see the comments): A001003 (Row1, Row2), A026003 (Kn1p, p >= 1), A006603 (Kn21), A227504 (Kn22), A227505 (Kn23), A006603(2*n) (Kn3), A001850 (Kn4), A227506 (Fi1), A010683 (Fi2).

a033877 n k = a033877_tabl !! n !! k
a033877_row n = a033877_tabl !! n
a033877_tabl = iterate
   (\row -> scanl1 (+) $ zipWith (+) ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Apr 17 2013

function t(n,k)
  if k le 0 or k gt n then return 0;
  elif k eq 1 then return 1;
  else return t(n,k-1) + t(n-1,k-1) + t(n-1,k);
  end if;
end function;
[t(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 23 2023

T := proc(n, k) option remember; if n=1 then return(1) fi; if kJohannes W. Meijer, Sep 22 2010, revised Jul 17 2013

Mathematica
```
T[1, ]:= 1; T[n, k_]/;(k
				
```

def A033877_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)-2*sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^k*prec(n, n-k) for k in (0..n-1)]
for n in (1..10): print(A033877_row(n)) # Peter Luschny, Mar 16 2016

@CachedFunction
def t(n, k): # t = A033847
    if (k<0 or k>n): return 0
    elif (k==1): return 1
    else: return t(n, k-1) + t(n-1, k-1) + t(n-1, k)
flatten([[t(n,k) for k in range(1,n+1)] for n in range(1, 16)]) # G. C. Greubel, Mar 23 2023

As an upper right triangle: a(n, k) = a(n, k-1) + a(n-1, k-1) + a(n-1, k) if k >= n >= 0 and a(n, k) = 0 otherwise.
G.f.: Sum T(n, k)*x^n*y^k = (1-x*y-(x^2*y^2-6*x*y+1)^(1/2)) / (x*(2*y+x*y-1+(x^2*y^2-6*x*y+1)^(1/2))). - Vladeta Jovovic, Feb 16 2003
Another version of A000007 DELTA [0, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] = 1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 16, 22, 0, 1, ..., where DELTA is Deléham's operator defined in A084938.
Sum_{n=1..floor((k+1)/2)} T(n+p-1, k-n+p) = A026003(2*p+k-3) - A008288(2*p+k-3, p-2), p >= 2, k >= 1. - Johannes W. Meijer, Sep 28 2013
From G. C. Greubel, Mar 23 2023: (Start)
(t(n, k) as a lower triangle)
t(n, k) = t(n, k-1) + t(n-1, k-1) + t(n-1, k) with t(n, 1) = 1.
t(n, n) = A006318(n-1).
t(2*n-1, n) = A330801(n-1).
t(2*n-2, n) = A103885(n-1), n > 1.
Sum_{k=1..n-1} t(n, k) = A238112(n), n > 1.
Sum_{k=1..n} t(n, k) = A001003(n).
Sum_{k=1..n-1} (-1)^(k-1)*t(n, k) = (-1)^n*A001003(n-1), n > 1.
Sum_{k=1..n} (-1)^(k-1)*t(n, k) = A080243(n-1).
Sum_{k=1..floor((n+1)/2)} t(n-k+1, k) = A026003(n-1). (End)

More terms from David W. Wilson

A113413 A Riordan array of coordination sequences.

Original entry on oeis.org

1, 2, 1, 2, 4, 1, 2, 8, 6, 1, 2, 12, 18, 8, 1, 2, 16, 38, 32, 10, 1, 2, 20, 66, 88, 50, 12, 1, 2, 24, 102, 192, 170, 72, 14, 1, 2, 28, 146, 360, 450, 292, 98, 16, 1, 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1, 2, 36, 258, 952, 1970, 2364, 1666, 688, 162, 20, 1, 2, 40, 326

Offset: 0

Paul Barry, Oct 29 2005

Columns include A040000, A008574, A005899, A008412, A008413, A008414. Row sums are A078057(n)=A001333(n+1). Diagonal sums are A001590(n+3). Reverse of A035607. Signed version is A080246. Inverse is A080245.
For another version see A122542. - Philippe Deléham, Oct 15 2006
T(n,k) is the number of length n words on alphabet {0,1,2} with no two consecutive 1's and no two consecutive 2's and having exactly k 0's. - Geoffrey Critzer, Jun 11 2015
From Eric W. Weisstein, Feb 17 2016: (Start)
Triangle of coefficients (from low to high degree) of x^-n * vertex cover polynomial of the n-ladder graph P_2 \square p_n:
  Psi_{L_1}: x*(2 + x) -> {2, 1}
  Psi_{L_2}: x^2*(2 + 4 x + x^2) -> {2, 4, 1}
  Psi_{L_3}: x^3*(2 + 8 x + 6 x^2 + x^3) -> {2, 8, 6, 1}
(End)
Let c(n, k), n > 0, be multiplicative sequences for some fixed integer k >= 0 with c(p^e, k) = T(e+k, k) for prime p and e >= 0. Then we have Dirichlet g.f.: Sum_{n>0} c(n, k) / n^s = zeta(s)^(2*k+2) / zeta(2*s)^(k+1). Examples: For k = 0 see A034444 and for k = 1 see A322328. Dirichlet convolution of c(n, k) and lambda(n) is Dirichlet inverse of c(n, k). - Werner Schulte, Oct 31 2022

			Triangle begins
  1;
  2,  1;
  2,  4,  1;
  2,  8,  6,  1;
  2, 12, 18,  8,  1;
  2, 16, 38, 32, 10,  1;
  2, 20, 66, 88, 50, 12,  1;

G. C. Greubel, Table of n, a(n) for the first 50 rows
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
P. Holub, M. Miller, H. Perez-Roses, and J. Ryan, Degree diameter problem on honeycomb networks, Disc. Appl. Math. 179 (2014) 139-151.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 2.
Mirka Miller, Hebert Perez-Roses, and Joe Ryan, The Maximum Degree-and-Diameter-Bounded Subgraph in the Mesh, arXiv preprint arXiv:1203.4069 [math.CO], 2012.

Other versions: A035607, A119800, A122542, A266213.

nn = 10; Map[Select[#, # > 0 &] &, CoefficientList[Series[1/(1 - 2 x/(1 + x) - y x), {x, 0, nn}], {x, y}]] // Grid (* Geoffrey Critzer, Jun 11 2015 *)
CoefficientList[CoefficientList[Series[1/(1 - 2 x/(1 + x) - y x), {x, 0, 10}, {y, 0, 10}], x], y] (* Eric W. Weisstein, Feb 17 2016 *)

T = lambda n,k : binomial(n, k)*hypergeometric([-k-1, k-n], [-n], -1).simplify_hypergeometric()
A113413 = lambda n,k : 1 if n==0 and k==0 else T(n, k)
for n in (0..12): print([A113413(n,k) for k in (0..n)]) # Peter Luschny, Sep 17 2014 and Mar 16 2016

# Alternatively:
def A113413_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)+2*sum(prec(n-i,k-1) for i in (2..n-k+1))
    return [prec(n, k) for k in (1..n)]
for n in (1..10): print(A113413_row(n)) # Peter Luschny, Mar 16 2016

From Paul Barry, Nov 13 2005: (Start)
Riordan array ((1+x)/(1-x), x(1+x)/(1-x)).
T(n, k) = Sum_{i=0..n-k} C(k+1, i)*C(n-i, k).
T(n, k) = Sum_{j=0..n-k} C(k+j, j)*C(k+1, n-k-j).
T(n, k) = D(n, k) + D(n-1, k) where D(n, k) = Sum_{j=0..n-k} C(n-k, j)*C(k, j)*2^j = A008288(n, k).
T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k-1).
T(n, k) = Sum_{j=0..n} C(floor((n+j)/2), k)*C(k, floor((n-j)/2)). (End)
T(n, k) = C(n, k)*hypergeometric([-k-1, k-n], [-n], -1). - Peter Luschny, Sep 17 2014
T(n, k) = (Sum_{i=2..k+2} A137513(k+2, i) * (n-k)^(i-2)) / (k!) for 0 <= k < n (conjectured). - Werner Schulte, Oct 31 2022

A080243 Signed super-Catalan or little Schroeder numbers.

Original entry on oeis.org

1, -1, 3, -11, 45, -197, 903, -4279, 20793, -103049, 518859, -2646723, 13648869, -71039373, 372693519, -1968801519, 10463578353, -55909013009, 300159426963, -1618362158587, 8759309660445, -47574827600981, 259215937709463, -1416461675464871, 7760733824437545, -42624971294485657

Offset: 0

Paul Barry, Feb 13 2003

sign

Row sums of triangle A080245.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, Exercise 2.7.12.(b).

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A001003, A080245.

CoefficientList[Series[(-1 + x + Sqrt[1 + 6 x + x^2]) /x / 4, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 05 2013 *)

x='x+O('x^66); Vec( (-1+x+sqrt(1+6*x+x^2))/x/4 ) \\ Joerg Arndt, Aug 15 2013

G.f.: (-1+x+sqrt(1+6*x+x^2))/x/4. - Vladeta Jovovic, Sep 27 2003
Conjecture: (n+1)*a(n) +3*(2*n-1)*a(n-1) +(n-2)*a(n-2)=0. - R. J. Mathar, Nov 26 2012
G.f.: 1 - x/(Q(0) + x) where Q(k) = 1 + k*(1+x) + x + x*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013
a(n) ~ (-1)^n * sqrt(4+3*sqrt(2)) * (3+2*sqrt(2))^n /(4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 15 2013
G.f. A(x) satisfies: A(x) = (1 - 2*x*A(x)^2) / (1 - x). - Ilya Gutkovskiy, Jun 30 2020

A080244 Signed generalized Fibonacci numbers.

Original entry on oeis.org

1, -2, 7, -26, 107, -468, 2141, -10124, 49101, -242934, 1221427, -6222838, 32056215, -166690696, 873798681, -4612654808, 24499322137, -130830894666, 702037771647, -3783431872018, 20469182526595, -111133368084892, 605312629105205, -3306633429423460, 18111655081108453

Offset: 1

Paul Barry, Feb 13 2003

Diagonal sums of triangle A080245

Vincenzo Librandi, Table of n, a(n) for n = 1..300

|a(n)| = A006603.

seq(coeff(convert(series((-1-x+2*x^2+sqrt(1+6*x+x^2))/(2*x*(1+x+x^2-x^3)),x,50),polynom),x,i),i=0..30); (C. Ronaldo)

CoefficientList[Series[(-1 - x + 2 x^2 + Sqrt[1 + 6 x + x^2]) / (2 x (1 + x + x^2 - x^3)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 05 2013 *)

G.f.: x*(-1-x+2*x^2+sqrt(1+6*x+x^2))/(2*x*(1+x+x^2-x^3)). - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 19 2004

Conjecture: (n+1)*a(n) +(7*n-2)*a(n-1) +4*(2*n-1)*a(n-2) +6*(n-1)*a(n-3) +(-5*n+1)*a(n-4) +(-n+2)*a(n-5)=0. - R. J. Mathar, Nov 24 2012

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 19 2004

G.f. adapted to the offset by Vincenzo Librandi, Aug 05 2013

A122538 Riordan array (1, x*f(x)) where f(x)is the g.f. of A006318.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 22, 16, 6, 1, 0, 90, 68, 30, 8, 1, 0, 394, 304, 146, 48, 10, 1, 0, 1806, 1412, 714, 264, 70, 12, 1, 0, 8558, 6752, 3534, 1408, 430, 96, 14, 1, 0, 41586, 33028, 17718, 7432, 2490, 652, 126, 16, 1, 0, 206098, 164512, 89898, 39152, 14002, 4080, 938, 160, 18, 1

Offset: 0

Philippe Deléham, Sep 18 2006

Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 2, 1, 2, 1, 2, 1, ...] DELTA [1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938 . Inverse is Riordan array (1, x*(1-x)/(1+x)).

T(n, r) gives the number of [0,r]-covering hierarchies with n segments terminating at r (see Kreweras work). - Michel Marcus, Nov 22 2014

			Triangle begins:
  1;
  0,    1:
  0,    2,    1;
  0,    6,    4,    1;
  0,   22,   16,    6,    1;
  0,   90,   68,   30,    8,   1;
  0,  394,  304,  146,   48,  10,  1;
  0, 1806, 1412,  714,  264,  70, 12,  1;
  0, 8558, 6752, 3534, 1408, 430, 96, 14, 1;
Production matrix is:
  0...1
  0...2...1
  0...2...2...1
  0...2...2...2...1
  0...2...2...2...2...1
  0...2...2...2...2...2...1
  0...2...2...2...2...2...2...1
  0...2...2...2...2...2...2...2...1
  0...2...2...2...2...2...2...2...2...1
  ... - _Philippe Deléham_, Feb 09 2014

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973), see page 15.

Another version : A080247, A080245, A033877.
Columns: A000007, A006318, A006319, A006320, A006321.
Diagonals: A000012, A005843, A054000.
Sums include: A001003 (row and alternating sign), A006603 (diagonal).
Cf. A103885.

function T(n,k) // T = A122538
  if k eq 0 then return 0^n;
  elif k eq n then return 1;
  else return T(n-1,k-1) + T(n-1,k) + T(n,k+1);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 27 2024

T[n_, n_]= 1; T[, 0]= 0; T[n, k_]:= T[n, k]= T[n-1, k-1] + T[n-1, k] + T[n, k+1];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, Jun 13 2019 *)

def A122538_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)-2*sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^(n-k)*prec(n, k) for k in (0..n)]
for n in (0..12): print(A122538_row(n)) # Peter Luschny, Mar 16 2016

T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n,k+1) if k > 0, with T(n, 0) = 0^n, and T(n, n) = 1.
Sum_{k=0..n} T(n, k) = A001003(n).
From G. C. Greubel, Oct 27 2024: (Start)
T(2*n, n) = A103885(n).
Sum_{k=0..n} (-1)^k*T(n, k) = -A001003(n-1).
Sum_{k=0..floor(n/2)} T(n-k, k) = [n=0] + 0*[n=1] + A006603(n-2)*[n>1]. (End)

cp's OEIS Frontend

A080247 Formal inverse of triangle A080246. Unsigned version of A080245.

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A033877 Triangular array read by rows associated with Schroeder numbers: T(1,k) = 1; T(n,k) = 0 if k < n; T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k).

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A113413 A Riordan array of coordination sequences.

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A080243 Signed super-Catalan or little Schroeder numbers.

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A080244 Signed generalized Fibonacci numbers.

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A122538 Riordan array (1, x*f(x)) where f(x)is the g.f. of A006318.

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