A002057 -id:A002057 - cp's OEIS frontend

A214292 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k), 0 < k < n with T(n,0) = n and T(n,n) = -n.

0, 1, -1, 2, 0, -2, 3, 2, -2, -3, 4, 5, 0, -5, -4, 5, 9, 5, -5, -9, -5, 6, 14, 14, 0, -14, -14, -6, 7, 20, 28, 14, -14, -28, -20, -7, 8, 27, 48, 42, 0, -42, -48, -27, -8, 9, 35, 75, 90, 42, -42, -90, -75, -35, -9, 10, 44, 110, 165, 132, 0, -132, -165, -110, -44, -10

Offset: 0

Reinhard Zumkeller, Jul 12 2012

			The triangle begins:
    0:                              0
    1:                            1   -1
    2:                          2   0   -2
    3:                       3    2   -2   -3
    4:                     4    5   0   -5   -4
    5:                  5    9    5   -5   -9   -5
    6:                6   14   14   0  -14  -14   -6
    7:             7   20   28   14  -14  -28  -20   -7
    8:           8   27   48   42   0  -42  -48  -27   -8
    9:        9   35   75   90   42  -42  -90  -75  -35   -9
   10:     10   44  110  165  132   0 -132 -165 -110  -44  -10
   11:  11   54  154  275  297  132 -132 -297 -275 -154  -54  -11  .

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A000004, A000096, A000108, A000245, A000344, A000588, A000589, A000590, A001392, A002057, A003517, A003518, A003519, A005557, A005586, A005587, A008313, A014495, A064059, A064061, A080956, A090749, A097808, A112467, A124087, A124088, A129936, A259525.

a214292 n k = a214292_tabl !! n !! k
a214292_row n = a214292_tabl !! n
a214292_tabl = map diff $ tail a007318_tabl
   where diff row = zipWith (-) (tail row) row

row[n_] := Table[Binomial[n, k], {k, 0, n}] // Differences;
T[n_, k_] := row[n + 1][[k + 1]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 31 2018 *)

T(n,k) = A007318(n+1,k+1) - A007318(n+1,k), 0<=k<=n, i.e. first differences of rows in Pascal's triangle;
T(n,k) = -T(n,k);
row sums and central terms equal 0, cf. A000004;
sum of positive elements of n-th row = A014495(n+1);
T(n,0) = n;
T(n,1) = A000096(n-2) for n > 1; T(n,1) = - A080956(n) for n > 0;
T(n,2) = A005586(n-4) for n > 3; T(n,2) = A129936(n-2);
T(n,3) = A005587(n-6) for n > 5;
T(n,4) = A005557(n-9) for n > 8;
T(n,5) = A064059(n-11) for n > 10;
T(n,6) = A064061(n-13) for n > 12;
T(n,7) = A124087(n) for n > 14;
T(n,8) = A124088(n) for n > 16;
T(2*n+1,n) = T(2*n+2,n) = A000108(n+1), Catalan numbers;
T(2*n+3,n) = A000245(n+2);
T(2*n+4,n) = A002057(n+1);
T(2*n+5,n) = A000344(n+3);
T(2*n+6,n) = A003517(n+3);
T(2*n+7,n) = A000588(n+4);
T(2*n+8,n) = A003518(n+4);
T(2*n+9,n) = A001392(n+5);
T(2*n+10,n) = A003519(n+5);
T(2*n+11,n) = A000589(n+6);
T(2*n+12,n) = A090749(n+6);
T(2*n+13,n) = A000590(n+7).

A059365 Another version of the Catalan triangle: T(r,s) = binomial(2r-s-1,r-1) - binomial(2r-s-1,r), r>=0, 0 <= s <= r.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 5, 3, 1, 0, 14, 14, 9, 4, 1, 0, 42, 42, 28, 14, 5, 1, 0, 132, 132, 90, 48, 20, 6, 1, 0, 429, 429, 297, 165, 75, 27, 7, 1, 0, 1430, 1430, 1001, 572, 275, 110, 35, 8, 1, 0, 4862, 4862, 3432, 2002, 1001, 429, 154, 44

Offset: 0

N. J. A. Sloane, Jan 28 2001

			Triangle starts
  0;
  0,    1;
  0,    1,    1;
  0,    2,    2,    1;
  0,    5,    5,    3,    1;
  0,   14,   14,    9,    4,    1;
  0,   42,   42,   28,   14,    5,   1;
  0,  132,  132,   90,   48,   20,   6,   1;
  0,  429,  429,  297,  165,   75,  27,   7,  1;
  0, 1430, 1430, 1001,  572,  275, 110,  35,  8, 1;
  0, 4862, 4862, 3432, 2002, 1001, 429, 154, 44, 9, 1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
D. Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], 2002.
FindStat - Combinatorial Statistic Finder, The number of touch points of a Dyck path., The number of initial rises of a Dyck paths., The number of nodes on the left branch of the tree., The number of subtrees.
A. Robertson, D. Saracino and D. Zeilberger, Refined restricted permutations, arXiv:math/0203033 [math.CO], 2002.
Index to sequences related to Catalan

See also the triangle in A009766. First 2 diagonals both give A000108, next give A000245, A002057.
Cf. A009766, A000007, A084938, A000108.
The three triangles A059365, A106566 and A099039 are the same except for signs and the leading term.
Essentially the same as A033184.
The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A053121, A059365, A099039, A106566, A130020, A047072, A171567, A181645.
Diagonals give A000108, A000245, A002057, A000344, A003517, A000588, A003518, A003519, A001392, ...

/* as triangle */ [[[0] cat [Binomial(2*r-s-1, r-1)- Binomial(2*r-s-1, r): s in [1..r]]: r in [0..10]]]; // Vincenzo Librandi, Jan 09 2017

Table[Binomial[2*r - s - 1, r - 1] - Binomial[2*r - s - 1, r], {r, 0, 10}, {s, 0, r}] // Flatten (* G. C. Greubel, Jan 08 2017 *)

tabl(nn) = { print(0); for (r=1, nn, for (s=0, r, print1(binomial(2*r-s-1,r-1)-binomial(2*r-s-1,r), ", ");); print(););}  \\ Michel Marcus, Nov 01 2013

Essentially the same triangle as [0, 1, 1, 1, 1, 1, 1, ...] DELTA A000007, where DELTA is Deléham's operator defined in A084938, but the first term is T(0,0) = 0.

A000588 a(n) = 7*binomial(2n,n-3)/(n+4).

Original entry on oeis.org

0, 0, 0, 1, 7, 35, 154, 637, 2548, 9996, 38760, 149226, 572033, 2187185, 8351070, 31865925, 121580760, 463991880, 1771605360, 6768687870, 25880277150, 99035193894, 379300783092, 1453986335186, 5578559816632, 21422369201800, 82336410323440, 316729578421620

Offset: 0

N. J. A. Sloane

a(n-5) is the number of n-th generation vertices in the tree of sequences with unit increase labeled by 6 (cf. Zoran Sunic reference). - Benoit Cloitre, Oct 07 2003
Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=3. Example: For n=3 there is only one path EEENNN. - Herbert Kociemba, May 24 2004
Number of standard tableaux of shape (n+3,n-3). - Emeric Deutsch, May 30 2004

			G.f. = x^3 + 7*x^4 + 35*x^5 + 154*x^6 + 637*x^7 + 2548*x^8 + 9996*x^9 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
Joerg Arndt, The a(5)=35 Young tableaux of shape [8,2].
Dennis E. Davenport, Louis W. Shapiro, Lara K. Pudwell and Leon C. Woodson, The Boundary of Ordered Trees, J. Integer Seq., Vol. 18 (2015), Article 15.5.8; alternative link.
Hilmar Haukur Gudmundsson, Dyck paths, standard Young tableaux, and pattern avoiding permutations, PU. M. A., Vol. 21, No. 2 (2010), pp. 265-284 (see 4.4 p. 279).
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., Vol. 14, No. 5 (1976), pp. 395-405.
Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math., Vol. 14 (1957), pp. 405-414. [Annotated scan of selected pages]
Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Applied Math., Vol. 14 (1956), pp. 405-414.
John Riordan, Letter to N. J. A. Sloane, Nov 10 1970.
John Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., Vol. 29, No. 129 (1975), pp. 215-222.
Zoran Sunic, Self-Describing Sequences and the Catalan Family Tree, Electronic Journal of Combinatorics, Vol. 10 (2003), Article N5.

First differences are in A026014.
A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Cf. A000108, A000245, A002057, A000344, A003517, A003518, A003519, A001392, A001622.

a[n_] := 7*Binomial[2n, n-3]/(n + 4); Table[a[n],{n,0,27}] (* James C. McMahon, Dec 05 2023 *)

A000588(n)=7*binomial(2*n,n-3)/(n+4) \\ M. F. Hasler, Aug 25 2012

my(x='x+O('x^50)); concat([0, 0, 0], Vec(x^3*((1-(1-4*x)^(1/2))/(2*x))^7)) \\ Altug Alkan, Nov 01 2015

Expansion of x^3*C^7, where C = (1-(1-4*x)^(1/2))/(2*x) is the g.f. for the Catalan numbers, A000108. - Philippe Deléham, Feb 03 2004
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=6, a(n-3)=(-1)^(n-6)*coeff(charpoly(A,x),x^6). - Milan Janjic, Jul 08 2010
a(n) = A214292(2*n-1,n-4) for n > 3. - Reinhard Zumkeller, Jul 12 2012
From Ilya Gutkovskiy, Jan 22 2017: (Start)
E.g.f.: (1/6)*x^3*1F1(7/2; 8; 4*x).
a(n) ~ 7*4^n/(sqrt(Pi)*n^(3/2)). (End)
0 = a(n)*(+1456*a(n+1) - 87310*a(n+2) + 132834*a(n+3) - 68068*a(n+4) + 9724*a(n+5)) + a(n+1)*(+8918*a(n+1) - 39623*a(n+2) + 51726*a(n+3) - 299*a(n+4) - 1573*a(n+5)) + a(n+2)*(-24696*a(n+2) - 1512*a(n+3) + 1008*a(n+4)) for all n in Z. - Michael Somos, Jan 22 2017
From Amiram Eldar, Jan 02 2022: (Start)
Sum_{n>=3} 1/a(n) = 27/14 - 26*Pi/(63*sqrt(3)).
Sum_{n>=3} (-1)^(n+1)/a(n) = 11364*log(phi)/(175*sqrt(5)) - 4583/350, where phi is the golden ratio (A001622). (End)
a(n) = Integral_{x=0..4} x^(n)*W(x)dx, n>=0, where W(x) = sqrt(4/x - 1)*(x^3 - 5*x^2 + 6*x - 1)/(2*Pi). The function W(x) for x->0 tends to -infinity (which is its absolute minimum), and W(4) = 0. W(x) is a signed function on the interval x = (0, 4) where it has two maxima separated by one local minimum. - Karol A. Penson, Jun 17 2024
D-finite with recurrence -(n+4)*(n-3)*a(n) +2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Jul 30 2024
a(n) = A000108(n+3) - 5*A000108(n+2) + 6*A000108(n+1) - A000108(n). - Taras Goy, Dec 21 2024

More terms from N. J. A. Sloane, Jul 13 2010

A001392 a(n) = 9*binomial(2n,n-4)/(n+5).

Original entry on oeis.org

1, 9, 54, 273, 1260, 5508, 23256, 95931, 389367, 1562275, 6216210, 24582285, 96768360, 379629720, 1485507600, 5801732460, 22626756594, 88152205554, 343176898988, 1335293573130, 5193831553416, 20198233818840, 78542105700240, 305417807763705

Offset: 4

N. J. A. Sloane

Number of n-th generation vertices in the tree of sequences with unit increase labeled by 8 (cf. Zoran Sunic reference) - Benoit Cloitre, Oct 07 2003
Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=4. - Herbert Kociemba, May 24 2004
Number of standard tableaux of shape (n+4,n-4). - Emeric Deutsch, May 30 2004

			G.f. = x^4 + 9*x^5 + 54*x^6 + 273*x^7 + 1260*x^8 + 5508*x^9 + 23256*x^10 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 4..200
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math., Vol. 14 (1957), pp. 405-414. [Annotated scan of selected pages]
Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Applied Math., Vol. 14 (1957), pp. 405-414.
John Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., Vol. 29, No. 129 (1975), pp. 215-222.
Zoran Sunic, Self-Describing Sequences and the Catalan Family Tree, Electronic Journal of Combinatorics, Vol. 10 (2003), Article #N5.

First differences are in A026015.
A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Cf. A000108, A000245, A002057, A000344, A003517, A000588, A003518, A003519, A001622.

A001392:=n->9*binomial(2*n,n-4)/(n+5): seq(A001392(n), n=4..40); # Wesley Ivan Hurt, Apr 11 2017

Table[9*Binomial[2n,n-4]/(n+5),{n,4,30}] (* Harvey P. Dale, Mar 03 2011 *)

a(n)=9*binomial(n+n,n-4)/(n+5) \\ Charles R Greathouse IV, Jul 31 2011

Expansion of x^4*C^9, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108. - Philippe Deléham, Feb 03 2004
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=8, a(n-4)=(-1)^(n-8)*coeff(charpoly(A,x),x^8). - Milan Janjic, Jul 08 2010
a(n) = A214292(2*n-1,n-5) for n > 4. - Reinhard Zumkeller, Jul 12 2012
D-finite with recurrence -(n+5)*(n-4)*a(n) +2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Jun 20 2013
From Ilya Gutkovskiy, Jan 22 2017: (Start)
E.g.f.: (1/24)*x^4*1F1(9/2; 10; 4*x).
a(n) ~ 9*4^n/(sqrt(Pi)*n^(3/2)). (End)
From Amiram Eldar, Jan 02 2022: (Start)
Sum_{n>=4} 1/a(n) = 158*Pi/(81*sqrt(3)) - 649/270.
Sum_{n>=4} (-1)^n/a(n) = 52076*log(phi)/(225*sqrt(5)) - 22007/450, where phi is the golden ratio (A001622). (End)

More terms from Harvey P. Dale, Mar 03 2011

A003518 a(n) = 8binomial(2n+1,n-3)/(n+5).

Original entry on oeis.org

1, 8, 44, 208, 910, 3808, 15504, 62016, 245157, 961400, 3749460, 14567280, 56448210, 218349120, 843621600, 3257112960, 12570420330, 48507033744, 187187399448, 722477682080, 2789279908316, 10772391370048, 41620603020640, 160878516023680, 622147386185325

Offset: 3

N. J. A. Sloane

a(n-6) is the number of n-th generation nodes in the tree of sequences with unit increase labeled by 7 (cf. Zoran Sunic reference). - Benoit Cloitre, Oct 07 2003

Number of standard tableaux of shape (n+4,n-3). - Emeric Deutsch, May 30 2004

			G.f. = x^3 + 8*x^4 + 44*x^5 + 208*x^6 + 910*x^7 + 3808*x^8 + 15504*x^9 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 3..500
Daniel Birmajer, Juan B. Gil and Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 9.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., Vol. 35, No. 4 (1995), pp. 743-751.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., Vol. 35, No. 4 (1995), pp. 743-751. [Annotated scanned copy]
Hilmar Haukur Gudmundsson, Dyck paths, standard Young tableaux, and pattern avoiding permutations, PU. M. A., Vol. 21, No.2 (2010), pp. 265-284; arXiv:0912.4747 [math.CO], 2009 (see Theorem 11 in Section 4.5).
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seq., Vol. 3 (2000), Article 00.1.6.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., Vol. 14, No. 5 (1976), pp. 395-405.
Olya Mandelshtam, Multi-Catalan tableaux and the two-species TASEP, arXiv:1502.00948 [math.CO], 2015.
Olya Mandelshtam, Multi-Catalan tableaux and the two-species TASEP, Ann. Inst. Henri Poincaré Comb. Phys. Interact., Vol. 3 (2016), pp. 321-348, DOI 10.4171/AIHPD/30.
L. W. Shapiro, A Catalan triangle, Discrete Math., Vol. 14, No. 1 (1976), pp. 83-90.
L. W. Shapiro, A Catalan triangle, Discrete Math., Vol. 14, No. 1 (1976), pp. 83-90. [Annotated scanned copy]
Zoran Sunic, Self describing sequences and the Catalan family tree, Elect. J. Combin., Vol. 10 (2003), Article N5.
Wen-Jin Woan, Lou Shapiro and D. G. Rogers, The Catalan numbers, the Lebesgue integral and 4^{n-2}, Amer. Math. Monthly, Vol. 104, No. 10 (1997), pp. 926-931.

Cf. A002057.
First differences are in A026018.
A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Cf. A000108, A000245, A000344, A000588, A001392, A002057, A003517, A003519, A001622.

[8*Binomial(2*n+1,n-3)/(n+5): n in [3..30]]; // Vincenzo Librandi, Jan 23 2017

Table[8 Binomial[2 n + 1, n - 3]/(n + 5), {n, 3, 25}] (* Michael De Vlieger, Oct 26 2016 *)
CoefficientList[Series[((1 - Sqrt[1 - 4 x])/(2 x))^8, {x, 0, 30}], x] (* Vincenzo Librandi, Jan 23 2017 *)

{a(n) = if( n<3, 0, 8 * binomial(2*n + 1, n-3) / (n + 5))}; /* Michael Somos, Mar 14 2011 */

my(x='x+O('x^50)); Vec(x^3*((1-(1-4*x)^(1/2))/(2*x))^8) \\ Altug Alkan, Nov 01 2015

G.f.: x^3*C(x)^8, where C(x)=(1-sqrt(1-4*x))/(2*x) is g.f. for the Catalan numbers (A000108). - Emeric Deutsch, May 30 2004
The convolution of A002057 with itself. - Gerald McGarvey, Nov 08 2007
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=7, a(n-4)=(-1)^(n-7)*coeff(charpoly(A,x),x^7). - Milan Janjic, Jul 08 2010
a(n) = A214292(2*n,n-4) for n > 3. - Reinhard Zumkeller, Jul 12 2012
Integral representation as the n-th moment of the signed weight function W(x) on (0,4), i.e.: a(n+3) = Integral_{x=0..4} x^n*W(x) dx, n >= 0, with W(x) = (1/2)*x^(7/2)*(x-2)*(x^2-4*x+2)*sqrt(4-x)/Pi. - Karol A. Penson, Oct 26 2016
From Ilya Gutkovskiy, Jan 22 2017: (Start)
E.g.f.: 4*BesselI(4,2*x)*exp(2*x)/x.
a(n) ~ 4^(n+2)/(sqrt(Pi)*n^(3/2)). (End)
D-finite with recurrence: -(n+5)*(n-3)*a(n) +2*n*(2*n+1)*a(n-1)=0. - R. J. Mathar, Feb 20 2020
From Amiram Eldar, Jan 02 2022: (Start)
Sum_{n>=3} 1/a(n) = 43*Pi/(36*sqrt(3)) - 81/80.
Sum_{n>=3} (-1)^(n+1)/a(n) = 6213*log(phi)/(50*sqrt(5)) - 10339/400, where phi is the golden ratio (A001622). (End)

More terms from Jon E. Schoenfield, May 06 2010

A030237 Catalan's triangle with right border removed (n > 0, 0 <= k < n).

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 1, 4, 9, 14, 1, 5, 14, 28, 42, 1, 6, 20, 48, 90, 132, 1, 7, 27, 75, 165, 297, 429, 1, 8, 35, 110, 275, 572, 1001, 1430, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 1, 10, 54, 208, 637, 1638, 3640, 7072, 11934, 16796, 1, 11, 65, 273, 910, 2548, 6188, 13260, 25194, 41990, 58786

Offset: 1

Wouter Meeussen

This triangle appears in the totally asymmetric exclusion process as Y(alpha=1,beta=1,n,m), written in the Derrida et al. reference as Y_n(m) for alpha=1, beta=1. - Wolfdieter Lang, Jan 13 2006

			Triangle begins as:
  1;
  1, 2;
  1, 3,  5;
  1, 4,  9,  14;
  1, 5, 14,  28,  42;
  1, 6, 20,  48,  90,  132;
  1, 7, 27,  75, 165,  297,  429;
  1, 8, 35, 110, 275,  572, 1001, 1430;
  1, 9, 44, 154, 429, 1001, 2002, 3432, 4862;

Reinhard Zumkeller, Rows n=1..151 of triangle, flattened
B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.
Wolfdieter Lang, First 10 rows.
Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015. See Fig. 6.

Alternate versions of (essentially) the same Catalan triangle: A009766, A033184, A047072, A059365, A099039, A106566, A130020.
Diagonals give A000108, A000245, A000344, A000588, A001392, A002057, A003517, A003518, A003519.
Row sums give A071724.
Cf. A009766, A350584.

a030237 n k = a030237_tabl !! n !! k
a030237_row n = a030237_tabl !! n
a030237_tabl = map init $ tail a009766_tabl
-- Reinhard Zumkeller, Jul 12 2012

[(n-k+1)*Binomial(n+k, k)/(n+1): k in [0..n-1], n in [1..12]]; // G. C. Greubel, Mar 17 2021

A030237 := proc(n,m)
    (n-m+1)*binomial(n+m,m)/(n+1) ;
end proc: # R. J. Mathar, May 31 2016
# Compare the analogue algorithm for the Bell numbers in A011971.
CatalanTriangle := proc(len) local P, T, n; P := [1]; T := [[1]];
for n from 1 to len-1 do P := ListTools:-PartialSums([op(P), P[-1]]);
T := [op(T), P] od; T end: CatalanTriangle(6):
ListTools:-Flatten(%); # Peter Luschny, Mar 26 2022
# Alternative:
ogf := n -> (1 - 2*x)/(1 - x)^(n + 2):
ser := n -> series(ogf(n), x, n):
row := n -> seq(coeff(ser(n), x, k), k = 0..n-1):
seq(row(n), n = 1..11); # Peter Luschny, Mar 27 2022

T[n_, k_]:= T[n, k] = Which[k==0, 1, k>n, 0, True, T[n-1, k] + T[n, k-1]];
Table[T[n, k], {n,1,12}, {k,0,n-1}] // Flatten (* Jean-François Alcover, Nov 14 2017 *)

T(n,k) = (n-k+1)*binomial(n+k, k)/(n+1) \\ Andrew Howroyd, Feb 23 2018

flatten([[(n-k+1)*binomial(n+k, k)/(n+1) for k in (0..n-1)] for n in (1..12)]) # G. C. Greubel, Mar 17 2021

T(n, k) = (n-k+1)*binomial(n+k, k)/(n+1).
Sum_{k=0..n-1} T(n,k) = A000245(n). - G. C. Greubel, Mar 17 2021
T(n, k) = [x^k] ((1 - 2*x)/(1 - x)^(n + 2)). - Peter Luschny, Mar 27 2022

Missing a(8) = T(7,0) = 1 inserted by Reinhard Zumkeller, Jul 12 2012

A000207 Number of inequivalent ways of dissecting a regular (n+2)-gon into n triangles by n-1 non-intersecting diagonals under rotations and reflections; also the number of (unlabeled) maximal outerplanar graphs on n+2 vertices.

Original entry on oeis.org

1, 1, 1, 3, 4, 12, 27, 82, 228, 733, 2282, 7528, 24834, 83898, 285357, 983244, 3412420, 11944614, 42080170, 149197152, 531883768, 1905930975, 6861221666, 24806004996, 90036148954, 327989004892, 1198854697588, 4395801203290, 16165198379984, 59609171366326, 220373278174641

Offset: 1

N. J. A. Sloane

Also a(n) is the number of hexaflexagons of order n+2. - Mike Godfrey (m.godfrey(AT)umist.ac.uk), Feb 25 2002 (see the Kosters paper).
Number of normally non-isomorphic realizations of the associahedron of type II with dimension n in Ceballos et al. - Tom Copeland, Oct 19 2011
Number of polyforms with n cells in the hyperbolic tiling with Schläfli symbol {3,oo}, not distinguishing enantiomorphs. - Thomas Anton, Jan 16 2019
A stereographic projection of the {3,oo} tiling on the Poincaré disk can be obtained via the Christensson link. - Robert A. Russell, Jan 20 2024
A maximal outerplanar graph (MOP) has a plane embedding with all vertices on the exterior region and interior regions triangles. - Allan Bickle, Feb 25 2024

			E.g., a square (4-gon, n=2) could have either diagonal drawn, C(3)=2, but with essentially only one result. A pentagon (5-gon, n=3) gives C(4)=5, but they each have 2 diags emanating from 1 of the 5 vertices and are essentially the same. A hexagon can have a nuclear disarmament sign (6 ways), an N (3 ways and 3 reflections) or a triangle (2 ways) of diagonals, 6 + 6 + 2 = 14 = C(5), but only 3 essentially different. - _R. K. Guy_, Mar 06 2004
G.f. = x + x^2 + x^3 + 3*x^4 + 4*x^5 + 12*x^6 + 27*x^7 + 82*x^8 + ...

L. W. Beineke and R. E. Pippert, Enumerating labeled k-dimensional trees and ball dissections, pp. 12-26 of Proceedings of Second Chapel Hill Conference on Combinatorial Mathematics and Its Applications, University of North Carolina, Chapel Hill, 1970. Reprinted in Math. Annalen, 191 (1971), 87-98.
Cameron, Peter J. Some treelike objects. Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155--183. MR0891613 (89a:05009). See pp. 155, 163, but note that the formulas on p. 163, lines 5 and 6, contain typos. See the correct formulas given here. - N. J. A. Sloane, Apr 18 2014
B. N. Cyvin, E. Brendsdal, J. Brunvoll and S. J. Cyvin, Isomers of polyenes attached to benzene, Croatica Chemica Acta, 68 (1995), 63-73.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.
R. K. Guy, "Dissecting a polygon into triangles," Bull. Malayan Math. Soc., Vol. 5, pp. 57-60, 1958.
R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 79, Table 3.5.1 (the entries for n=16 and n=21 appear to be incorrect).
M. Kosters, A theory of hexaflexagons, Nieuw Archief Wisk., 17 (1999), 349-362.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. K. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974.

T. D. Noe, Table of n, a(n) for n = 1..200
F. R. Bernhart & N. J. A. Sloane, Correspondence, 1977
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Douglas Bowman and Alon Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv preprint arXiv:1209.6270 [math.CO], 2012.
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183. See p. 160.
C. Ceballos, F. Santos, and G. Ziegler, Many Non-equivalent Realizations of the Associahedron, arXiv:1109.5544 [math.MG], 2011-2013, p. 19 and 26.
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
Sean Cleary, Roland Maio, Counting difficult tree pairs with respect to the rotation distance problem, arXiv:2001.06407 [cs.DS], 2020.
A. S. Conrad and D. K. Hartline, Flexagons
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]
R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967. [Annotated scanned copy]
F. Harary and E. M. Palmer, On acyclic simplicial complexes, Mathematika 15 1968 115-122.
F. Harary, E. M. Palmer, and R. C. Read, On the cell-growth problem for arbitrary polygons, computer printout, circa 1974
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389 (the entries for n=4 and n=30 appear to be incorrect).
J. W. Moon and L. Moser, Triangular dissections of n-gons, Canad. Math. Bull., 6 (1963), 175-178.
T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51 (1945), 976-984.
T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360 (the entry for n=10 appears to be incorrect).
C. O. Oakley and R. J. Wisner, Flexagons, Amer. Math. Monthly 64 (1957), 143-154.
Hans Rademacher, On the number of certain types of polyhedra, Illinois Journal of Mathematics 9.3 (1965): 361-380. Reprinted in Coll. Papers, Vol II, MIT Press, 1974, pp. 544-564.
Manfred Scheucher, Hendrik Schrezenmaier, Raphael Steiner, A Note On Universal Point Sets for Planar Graphs, arXiv:1811.06482 [math.CO], 2018.
Len Smiley, Illustration of initial terms
Tiberiu Spircu and Stefan V. Pantazi, Again around frieze patterns, arXiv:2002.08211 [math.CO], 2020. See Kn p. 13.
P. J. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974. [Scanned annotated and corrected copy]

Column k=3 of A295260.
Cf. A000577, A070765.
A row or column of the array in A169808.
Polyominoes: A001683(n+2) (oriented), A369314 (chiral), A208355(n-1) (achiral), A005036 {4,oo}, A007173 {3,3,oo}.
Cf. A097998, A097999, A098000 (labeled outerplanar graphs).
Cf. A111563, A111564, A111758, A111759, A111757 (unlabeled outerplanar graphs).

A000108 := proc(n) if n >= 0 then binomial(2*n,n)/(n+1) ; else 0; fi; end:
A000207 := proc(n) option remember: local k, it1, it2;
if n mod 2 = 0 then k := n/2+2 else k := (n+3)/2 fi:
if n mod 2 <> 0 then it1 := 0 else it1 := 1 fi:
if (n+2) mod 3 <> 0 then it2 := 0 else it2 := 1 fi:
RETURN(A000108(n)/(2*n+4) + it1*A000108(n/2)/4 + A000108(k-2)/2 + it2*A000108((n-1)/3)/3)
end:
seq(A000207(n),n=1..30) ; # (Revised Maple program from R. J. Mathar, Apr 19 2009)
A000207 := proc(n) option remember: local k,it1,it2; if n mod 2 = 0 then k := n/2+1 else k := (n+1)/2 fi: if n mod 2 <> 0 then it1 := 0 else it1 := 1 fi: if n mod 3 <> 0 then it2 := 0 else it2 := 1 fi: RETURN(A000108(n-2)/(2*n) + it1*A000108(n/2+1-2)/4 + A000108(k-2)/2 + it2*A000108(n/3+1-2)/3) end:
A000207 := n->(A000108(n)/(n+2)+A000108(floor(n/2))*((1+(n+1 mod 2) /2)))/2+`if`(n mod 3=1,A000108(floor((n-1)/3))/3,0); # Peter Luschny, Apr 19 2009 and M. F. Hasler, Apr 19 2009
G:=(12*(1+x-2*x^2)+(1-4*x)^(3/2)-3*(3+2*x)*(1-4*x^2)^(1/2)-4*(1-4*x^3)^(1/2))/24/x^2: Gser:=series(G,x=0,35): seq(coeff(Gser,x^n),n=1..31); # Emeric Deutsch, Dec 19 2004

p=3; Table[(Binomial[(p-1)n, n]/(((p-2)n+1)((p-2)n+2)) + If[OddQ[n], If[OddQ[p], Binomial[(p-1)n/2, (n-1)/2]/n, (p+1)Binomial[((p-1)n-1)/2, (n-1)/2]/((p-2)n+2)], 3Binomial[(p-1)n/2, n/2]/((p-2)n+2)]+Plus @@ Map[EulerPhi[ # ]Binomial[((p-1)n+1)/#, (n-1)/# ]/((p-1)n+1)&, Complement[Divisors[GCD[p, n-1]], {1, 2}]])/2, {n, 1, 20}] (* Robert A. Russell, Dec 11 2004 *)
a[n_] := (CatalanNumber[n]/(n+2) + CatalanNumber[ Quotient[n, 2]] *((1 + Mod[n-1, 2]/2)))/2 + If[Mod[n, 3] == 1, CatalanNumber[ Quotient[n-1, 3]]/3, 0] ; Table[a[n], {n, 1, 28}] (* Jean-François Alcover, Sep 08 2011, after PARI *)

A000207(n)=(A000108(n)/(n+2)+A000108(n\2)*if(n%2,1,3/2))/2+if(n%3==1,A000108(n\3)/3) \\ M. F. Hasler, Apr 19 2009

a(n) = C(n)/(2*n) + C(n/2+1)/4 + C(k)/2 + C(n/3+1)/3 where C(n) = A000108(n-2) if n is an integer, 0 otherwise and k = (n+1)/2 if n is odd, k = n/2+1 if n is even. Thus C(2), C(3), C(4), C(5), ... are 1, 1, 2, 5, ...
G.f.: (12*(1+x-2*x^2) + (1-4*x)^(3/2) - 3*(3+2*x)*(1-4*x^2)^(1/2) - 4*(1-4*x^3)^(1/2))/(24*x^2). - Emeric Deutsch, Dec 19 2004, from the S. J. Cyvin et al. reference.
a(n) ~ A000108(n)/(2*n+4) ~ 4^n / (2 sqrt(n Pi)*(n + 1)*(n + 2)). - M. F. Hasler, Apr 19 2009
a(n) = A001683(n+2) - A369314(n) = (A001683(n+2) + A208355(n-1)) / 2 = A369314(n) + A208355(n-1). - Robert A. Russell, Jan 19 2024
Beineke and Pippert have an explicit formula with six cases (based on the value of n mod 6). - Allan Bickle, Feb 25 2024

More terms from James Sellers, Jul 10 2000

A003519 a(n) = 10*C(2n+1, n-4)/(n+6).

Original entry on oeis.org

1, 10, 65, 350, 1700, 7752, 33915, 144210, 600875, 2466750, 10015005, 40320150, 161280600, 641886000, 2544619500, 10056336264, 39645171810, 155989499540, 612815891050, 2404551645100, 9425842448792, 36921502679600, 144539291740025, 565588532895750, 2212449261033375

Offset: 4

N. J. A. Sloane

Number of standard tableaux of shape (n+5,n-4). - Emeric Deutsch, May 30 2004

a(n) is the number of North-East paths from (0,0) to (n,n) that cross the diagonal y = x horizontally exactly twice. By symmetry, it is also the number of North-East paths from (0,0) to (n,n) that cross the diagonal y = x vertically exactly twice. Details can be found in Section 3.3 in Pan and Remmel's link. - Ran Pan, Feb 02 2016

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Israel, Table of n, a(n) for n = 4..1650
Daniel Birmajer, Juan B. Gil and Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 9.
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., Vol. 14, No. 5 (1976), pp. 395-405.
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.

Cf. A000108, A000245, A002057, A000344, A003517, A000588, A003518, A001392, A001622.

[10*Binomial(2*n+1, n-4)/(n+6): n in [4..35]]; // Vincenzo Librandi, Feb 03 2016

seq(10*binomial(2*n+1,n-4)/(n+6), n=4..50); # Robert Israel, Feb 02 2016

Table[10 Binomial[2 n + 1, n - 4]/(n + 6), {n, 4, 28}] (* Michael De Vlieger, Feb 03 2016 *)

a(n) = 10*binomial(2*n+1, n-4)/(n+6); \\ Michel Marcus, Feb 02 2016

G.f.: x^4*C(x)^10, where C(x)=[1-sqrt(1-4x)]/(2x) is g.f. for the Catalan numbers (A000108). - Emeric Deutsch, May 30 2004
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=9, a(n-5)=(-1)^(n-9)*coeff(charpoly(A,x),x^9). [Milan Janjic, Jul 08 2010]
a(n) = A214292(2*n,n-5) for n > 4. - Reinhard Zumkeller, Jul 12 2012
From Robert Israel, Feb 02 2016: (Start)
D-finite with recurrence a(n+1) = 2*(n+1)*(2n+3)/((n+7)*(n-3)) * a(n).
a(n) ~ 20 * 4^n/sqrt(Pi*n^3). (End)
E.g.f.: 5*BesselI(5,2*x)*exp(2*x)/x. - Ilya Gutkovskiy, Jan 23 2017
From Amiram Eldar, Jan 02 2022: (Start)
Sum_{n>=4} 1/a(n) = 34*Pi/(45*sqrt(3)) - 44/175.
Sum_{n>=4} (-1)^n/a(n) = 53004*log(phi)/(125*sqrt(5)) - 79048/875, where phi is the golden ratio (A001622). (End)

A068875 Expansion of (1 + xC)C, where C = (1 - (1 - 4x)^(1/2))/(2x) is the g.f. for Catalan numbers, A000108.

Original entry on oeis.org

1, 2, 4, 10, 28, 84, 264, 858, 2860, 9724, 33592, 117572, 416024, 1485800, 5348880, 19389690, 70715340, 259289580, 955277400, 3534526380, 13128240840, 48932534040, 182965127280, 686119227300, 2579808294648, 9723892802904, 36734706144304, 139067101832008

Offset: 0

N. J. A. Sloane, Jun 06 2002

A Catalan transform of A040000 under the mapping g(x) -> g(x*c(x)), where c(x) is the g.f. of A000108. Sequence A040000 can be retrieved using the mapping g(x) -> g(x*(1-x)). A040000(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k) * (-1)^k * a(n-k). a(n) and A040000 may be described as a Catalan pair. - Paul Barry, Nov 14 2004
a(n) is the number of Dyck (n+1)-paths all of whose nonterminal descents to ground level are of odd length. For example, a(2) counts UUUDDD, UUDUDD, UDUUDD, UDUDUD. - David Callan, Jul 25 2005
From Gary W. Adamson, Jul 11 2011: (Start)
a(n) is the sum of the top row terms in M^n, where M is the following infinite square production matrix:
  1, 1, 0, 0, 0, 0, ...
  0, 1, 1, 0, 0, 0, ...
  1, 1, 1, 1, 0, 0, ...
  0, 1, 1, 1, 1, 0, ...
  1, 1, 1, 1, 1, 1, ...
  ...
For example, the top row of M^3 = (2, 4, 3, 1) with sum = 10 = a(3). (End)
For n >= 1, a(n) is the number of binary trees with n+1 internal node in which one of the subtrees of the root is empty. Cf. A002057. [Sedgewick and Flajolet] - Geoffrey Critzer, Jan 05 2013
Empirical: a(n) is the number of entries of absolute value 1 that appear among all partitions in the canonical basis of the Temperley-Lieb algebra of order n. - John M. Campbell, Oct 17 2017
For n >= 1, a(n) is the number of Dyck paths of size n+2, whose corresponding unit interval graph has P3-hull number equal to 2. This result is due to Alrik Sandberg. - Per W. Alexandersson, Jan 09 2024

			G.f. = 1 + 2*x + 4*x^2 + 10*x^3 + 28*x^4 + 84*x^5 + 264*x^6 + 858*x^7 + ...
For example, the canonical basis of the Temperley-Lieb algebra of order 3 is {{{-3, 1}, {-2, -1}, {2, 3}}, {{-3, 3}, {-2, 2}, {-1, 1}}, {{-3, 3}, {-2, -1}, {1, 2}}, {{-3, -2}, {-1, 1}, {2, 3}}, {{-3, -2}, {-1, 3}, {1, 2}}}, and we see that the total number of entries of absolute value 1 that appear among the partitions in this basis is a(3) = 10.

R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley, 1996, p. 225.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995), 743-751. [Annotated scanned copy]
S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, 2017.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

A002420 and A262543 are essentially the same sequence as this.

Cf. A000108, A000984, A002057, A040000, A068875, A071721.

[1] cat [2*Binomial( 2*n, n)/(n+1): n in [1..30]]; // Vincenzo Librandi, Oct 17 2017

Z:=(1-sqrt(1-4*x))/2/x: G:=(2-(1+x)*Z)/Z: Gser:=series(-G, x=0, 30): (1,seq(coeff(Gser, x, n), n=2..26)); # Zerinvary Lajos, Dec 23 2006
Z:=-(1-z-sqrt(1-z))/sqrt(1-z): Zser:=series(Z, z=0, 32): (1,seq(coeff(Zser*4^n, z, n), n=2..26)); # Zerinvary Lajos, Jan 01 2007
A068875List := proc(m) local A, P, n; A := [1, 2]; P := [2];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A068875List(26); # Peter Luschny, Mar 24 2022

nn=30;t=(1-(1-4x )^(1/2))/(2x);Prepend[Table[Coefficient[Series[1+x (y t -y+1)^2,{x,0,nn}],x ^n y],{n,2,nn}],1]  (* Geoffrey Critzer, Jan 05 2013 *)
a[ n_] := If[ n < 1, Boole[ n == 0], 2 Binomial[ 2 n, n]/(n + 1)]; (* Michael Somos, Jun 18 2014 *)
a[ n_] := SeriesCoefficient[ -1 + 4 / (1 + Sqrt[ 1 - 4 x]), {x, 0, n}]; (* Michael Somos, Jun 18 2014 *)
Table[If[n==0, 1, 2 CatalanNumber[n]], {n,0,25}] (* Peter Luschny, Feb 27 2017 *)

{a(n) = if( n<1, n==0, 2 * binomial( 2*n, n) / (n + 1))}; /* Michael Somos, Aug 17 2005 */

{a(n) = if( n<0, 0, polcoeff( -1 + 4 / (1 + sqrt(1 - 4*x + x * O(x^n))), n))}; /* Michael Somos, Aug 17 2005 */

Apart from initial term, twice Catalan numbers.
G.f.: (1 - x - sqrt(1 - 4*x)) / x. - Michael Somos, Apr 13 2012
From Paul Barry, Nov 14 2004: (Start)
G.f.: (1 + x*c(x))/(1 - x*c(x)), where c(x) is the g.f. of A000108.
a(n) = C(n)*(2-0^n), where C(n) = A000108(n).
a(n) = Sum_{j=0..n} Sum_{k=0..n} binomial(2*n, n-k) *((2*k + 1)/(n + k + 1)) * binomial(k, j) * (-1)^(j-k) * (2 - 0^j). (End)
Assuming offset 1, then series reversion of g.f. A(x) is -A(-x). - Michael Somos, Aug 17 2005
Assuming offset 2, then A(x) satisfies A(x - x^2) = x^2 - x^4 and so A(x) = C(x)^2 - C(x)^4, A(A(x)) = C(x)^4 - C(x)^8, A(A(A(x))) = C(x)^8 - C(x)^16, etc., where C(x) = (1-sqrt(1-4*x))/2 = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + ... . - Paul D. Hanna, May 16 2008
Apart from initial term, INVERTi transform of A000984(n+1) = binomial(2*n+2,n+1), also, for n >= 1, a(n) = (1/Pi)*Integral_{x=0..4} x^(n-1)*sqrt(x*(4 - x)). - Groux Roland, Mar 15 2011
D-finite with recurrence (n+2)*a(n) - 2*(2*n+1)*a(n-1) = 0, n > 1. - R. J. Mathar, Nov 14 2011
For n > 0, a(n) = C(2*n+2, n+1) mod 4*C(2*n, n - 1). - Robert G. Wilson v, May 02 2012
For n > 0, a(n) = 2^(2*n + 1) * Gamma(n + 1/2)/(sqrt(Pi) * (n + 1)!). - Vaclav Kotesovec, Sep 16 2013
G.f.: 1 + 2*x/(Q(0) - x), where Q(k) = 2*x + (k + 1)/(2*k + 1) - 2*x*(k + 1)/(2*k + 1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 03 2013
G.f.: 3 - 4*x - 2*S(0), where S(k) = 2*k + 1 - x*(2*k + 3)/(1 - x*(2*k + 1)/S(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 23 2013
0 = a(n)*(16*a(n+1) - 10*a(n+2)) + a(n+1)*(2*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Jun 18 2014
If A(x)^t = 1 + 2*t*x + Sum_{n >= 2} t*P(n,t)*x^n, then we conjecture that all the zeros of the polynomial P(n,t) lie on the vertical line Re(t) = -n/2 in the complex plane. - Peter Bala, Oct 05 2015
a(n+1) = a(n) + (1/2)*(Sum_{k=0..n} a(k)*a(n-k)) if n > 0. - Michael Somos, Apr 22 2022
b(n) = a(n+1) - a(n) for all n in Z if b(0) = 2, a(-1) = -1, a(0) = 0, a(-1) = 3, a(-2) = -1 where b = A071721. - Michael Somos, Apr 23 2022

A047072 Array A read by diagonals: A(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and no step touches the line y=x unless x=0 or x=h.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 2, 2, 3, 1, 1, 4, 5, 4, 5, 4, 1, 1, 5, 9, 5, 5, 9, 5, 1, 1, 6, 14, 14, 10, 14, 14, 6, 1, 1, 7, 20, 28, 14, 14, 28, 20, 7, 1, 1, 8, 27, 48, 42, 28, 42, 48, 27, 8, 1, 1, 9, 35, 75, 90, 42, 42, 90, 75, 35, 9, 1

Offset: 0

Clark Kimberling

			Array, A(n, k), begins as:
  1, 1,  1,  1,  1,   1,   1,   1, ...;
  1, 2,  1,  2,  3,   4,   5,   6, ...;
  1, 1,  2,  2,  5,   9,  14,  20, ...;
  1, 2,  2,  4,  5,  14,  28,  48, ...;
  1, 3,  5,  5, 10,  14,  42,  90, ...;
  1, 4,  9, 14, 14,  28,  42, 132, ...;
  1, 5, 14, 28, 42,  42,  84, 132, ...;
  1, 6, 20, 48, 90, 132, 132, 264, ...;
Antidiagonals, T(n, k), begins as:
  1;
  1,  1;
  1,  2,  1;
  1,  1,  1,  1;
  1,  2,  2,  2,  1;
  1,  3,  2,  2,  3,  1;
  1,  4,  5,  4,  5,  4,  1;
  1,  5,  9,  5,  5,  9,  5,  1;
  1,  6, 14, 14, 10, 14, 14,  6,  1;

G. C. Greubel, Antidiagonals n = 0..50, flattened
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

Cf. A000007, A000027, A000096, A002420.
Cf. A005586, A025174, A047079, A063886.
Cf. A047073, A047074, A047079.
The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Diagonals give A000108, A000245, A002057, A000344, A003517, A000588, A003518, A003519, A001392, ...

b:= func< n | n eq 0 select 1 else 2*Catalan(n-1) >;
function A(n,k)
  if k eq n then return b(n);
  elif k gt n then return Binomial(n+k-1, n) - Binomial(n+k-1, n-1);
  else return Binomial(n+k-1, k) - Binomial(n+k-1, k-1);
  end if; return A;
end function;
// [[A(n,k): k in [0..12]]: n in [0..12]];
T:= func< n,k | A(n-k, k) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 13 2022

A[, 0]= 1; A[0, ]= 1; A[h_, k_]:= A[h, k]= If[(k-1>h || k-1Jean-François Alcover, Mar 06 2019 *)

def A(n,k):
    if (k==n): return 2*catalan_number(n-1) + 2*int(n==0)
    elif (k>n): return binomial(n+k-1, n) - binomial(n+k-1, n-1)
    else: return binomial(n+k-1, k) - binomial(n+k-1, k-1)
def T(n,k): return A(n-k, k)
# [[A(n,k) for k in range(12)] for n in range(12)]
flatten([[T(n,k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Oct 13 2022

A(n, n) = 2*[n=0] - A002420(n),
A(n, n+1) = 2*A000108(n-1), n >= 1.
From G. C. Greubel, Oct 13 2022: (Start)
T(n, n-1) = A000027(n-2) + 2*[n<3], n >= 1.
T(n, n-2) = A000096(n-4) + 2*[n<5], n >= 2.
T(n, n-3) = A005586(n-6) + 4*[n<7] - 2*[n=3], n >= 3.
T(2*n, n) = 2*A000108(n-1) + 3*[n=0].
T(2*n-1, n-1) = T(2*n+1, n+1) = A000180(n).
T(3*n, n) = A025174(n) + [n=0]
Sum_{k=0..n} T(n, k) = 2*A063886(n-2) + [n=0] - 2*[n=1]
Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n, k) = A047079(n). (End)

cp's OEIS Frontend

A214292 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k), 0 < k < n with T(n,0) = n and T(n,n) = -n.

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A059365 Another version of the Catalan triangle: T(r,s) = binomial(2*r-s-1,r-1) - binomial(2*r-s-1,r), r>=0, 0 <= s <= r.

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A000588 a(n) = 7*binomial(2n,n-3)/(n+4).

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A001392 a(n) = 9*binomial(2n,n-4)/(n+5).

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A003518 a(n) = 8*binomial(2*n+1,n-3)/(n+5).

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A030237 Catalan's triangle with right border removed (n > 0, 0 <= k < n).

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A059365 Another version of the Catalan triangle: T(r,s) = binomial(2r-s-1,r-1) - binomial(2r-s-1,r), r>=0, 0 <= s <= r.

A003518 a(n) = 8binomial(2n+1,n-3)/(n+5).

A068875 Expansion of (1 + xC)C, where C = (1 - (1 - 4x)^(1/2))/(2x) is the g.f. for Catalan numbers, A000108.