A003518 -id:A003518 - cp's OEIS frontend

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

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

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

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)

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)

A130020 Triangle T(n,k), 0<=k<=n, read by rows given by [1,0,0,0,0,0,0,...] DELTA [0,1,1,1,1,1,1,...] where DELTA is the operator defined in A084938 .

Original entry on oeis.org

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

Offset: 0

Philippe Deléham, Jun 16 2007

Reflected version of A106566.

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

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Francesca Aicardi, Catalan triangle and tied arc diagrams, arXiv:2011.14628 [math.CO], 2020.

The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A047072, A059365, A099039, A106566, this sequence.
Diagonals give A000108, A000245, A002057, A000344, A003517, A000588, A003518, A003519, A001392, ...
Cf. A000108 (Catalan numbers), A106566 (row reversal), A210736.

A130020:= func< n,k | n eq 0 select 1 else (n-k)*Binomial(n+k-1, k)/n >;
[A130020(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 14 2022

T[n_, k_]:= (n-k)Binomial[n+k-1, k]/n; T[0, 0] = 1;
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Jun 14 2019 *)

{T(n, k) = if( k<0 || k>=n, n==0 && k==0, binomial(n+k, n) * (n-k)/(n+k))}; /* Michael Somos, Oct 01 2022 */

@CachedFunction
def A130020(n, k):
    if n==k: return add((-1)^j*binomial(n, j) for j in (0..n))
    return add(A130020(n-1, j) for j in (0..k))
for n in (0..10) :
    [A130020(n, k) for k in (0..n)]  # Peter Luschny, Nov 14 2012

T(n, k) = A106566(n, n-k).
Sum_{k=0..n} T(n,k) = A000108(n).
T(n, k) = (n-k)*binomial(n+k-1, k)/n with T(0, 0) = 1. - Jean-François Alcover, Jun 14 2019
Sum_{k=0..floor(n/2)} T(n-k, k) = A210736(n). - G. C. Greubel, Jun 14 2022
G.f.: Sum_{n>=0, k>=0} T(n, k)*x^k*z^n = 1/(1 - z*c(x*z)) where c(z) = g.f. of A000108.

A099039 Riordan array (1,c(-x)), where c(x) = g.f. of Catalan numbers.

Original entry on oeis.org

1, 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, 0, 16796, -16796, 11934, -7072, 3640, -1638

Offset: 0

Paul Barry, Sep 23 2004

Row sums are generalized Catalan numbers A064310. Diagonal sums are 0^n+(-1)^n*A030238(n-2). Inverse is A026729, as number triangle. Columns have g.f. (xc(-x))^k=((sqrt(1+4x)-1)/2)^k.

Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, ... ] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... ] where DELTA is the operator defined in A084938. - Philippe Deléham, May 31 2005

			Rows begin {1}, {0,1}, {0,-1,1}, {0,2,-2,1}, {0,-5,5,-3,1}, ...
Triangle begins
  1;
  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;
Production matrix is
  0,  1,
  0, -1,  1,
  0,  1, -1,  1,
  0, -1,  1, -1,  1,
  0,  1, -1,  1, -1,  1,
  0, -1,  1, -1,  1, -1,  1,
  0,  1, -1,  1, -1,  1, -1,  1,
  0, -1,  1, -1,  1, -1,  1, -1,  1,
  0,  1, -1,  1, -1,  1, -1,  1, -1,  1

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021. See (2.10) p. 6.
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.
E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
A. Robertson, D. Saracino and D. Zeilberger, Refined restricted permutations, arXiv:math/0203033 [math.CO], 2002.
L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.

The three triangles A059365, A106566 and A099039 are the same except for signs and the leading term.
Cf. A033184, A000108.
Cf. A106566 (unsigned version), A059365
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, ...

T[n_, k_]:= If[n == 0 && k == 0, 1, If[n == 0 && k > 0, 0, (-1)^(n + k)*Binomial[2*n - k - 1, n - k]*k/n]];  Table[T[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* G. C. Greubel, Dec 31 2017 *)

{T(n,k) = if(n == 0 && k == 0, 1, if(n == 0 && k > 0, 0, (-1)^(n + k)*binomial(2*n - k - 1, n - k)*k/n))};
for(n=0,15, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 31 2017

T(n, k) = (-1)^(n+k)*binomial(2*n-k-1, n-k)*k/n for 0 <= k <= n with n > 0; T(0, 0) = 1; T(0, k) = 0 if k > 0. - Philippe Deléham, May 31 2005

A050144 T(n,k) = M(2n-1,n-1,k-1), 0 <= k <= n, n >= 0, where M(p,q,r) is the number of upright paths from (0,0) to (p,p-q) that meet the line y = x+r and do not rise above it.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 2, 3, 4, 1, 5, 9, 14, 6, 1, 14, 28, 48, 27, 8, 1, 42, 90, 165, 110, 44, 10, 1, 132, 297, 572, 429, 208, 65, 12, 1, 429, 1001, 2002, 1638, 910, 350, 90, 14, 1, 1430, 3432, 7072, 6188, 3808, 1700, 544, 119, 16, 1

Offset: 0

Clark Kimberling

Let V=(e(1),...,e(n)) consist of q 1's and p-q 0's; let V(h)=(e(1),...,e(h)) and m(h)=(#1's in V(h))-(#0's in V(h)) for h=1,...,n. Then M(p,q,r)=number of V having r=max{m(h)}.
The interpretation of T(n,k) as RU walks in terms of M(.,.,.) in the NAME is erroneous. There seems to be a pattern along subdiagonals:
 M(3,1,1) = 4  = T(3,2); M(3,1,2) = 1  = T(4,4); M(5,2,1) = 20 = T(5,3); M(5,2,2) = 7  = T(6,5); M(5,2,3) = 1  = T(7,7); M(7,3,0) = 165 = T(6,2); M(7,3,1) = 110 = T(7,4); M(7,3,2) = 44  = T(8,6); M(7,3,3) = 10  = T(9,8); M(7,3,4) = 1   = T(10,10); M(9,4,0) = 1001 = T(8,3); M(9,4,1) = 637  = T(9,5); M(9,4,2) = 273  = T(10,7); M(9,4,3) = 77   = T(11,9); M(9,4,4) = 13   = T(12,11); M(9,4,5) = 1   = T(13,13); M(11,5,0) = 6188 = T(10,4); M(11,5,1) = 3808 = T(11,6); M(11,5,2) = 1700 = T(12,8); M(11,5,3) = 544  = T(13,...); M(11,5,4) = 119; M(11,5,5) = 16; M(11,5,6) = 1; M(13,6,0) = 38760 = T(12,5); M(13,6,1) = 23256 = T(13,7); M(13,6,2) = 10659 = T(14,9); - R. J. Mathar, Jul 31 2024

			Triangle begins:
     0
     1    0
     1    1    1
     2    3    4    1
     5    9   14    6    1
    14   28   48   27    8    1
    42   90  165  110   44   10    1
   132  297  572  429  208   65   12    1
   429 1001 2002 1638  910  350   90   14    1
  1430 3432 7072 6188 3808 1700  544  119   16    1

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Emeric Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 29.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

{M(2n, 0, k)} is given by A039599. {M(2n+1, n+1, k+1)} is given by A039598.

Cf. A033184, A050153, A000108 (column 0), A000245 (column 1), A002057 (column 2), A000344 (column 3), A003517 (column 4), A000588 (column 5), A003518 (column 6), A001392 (column 7), A003519 (column 8), A000589 (column 9), A090749 (column 10).

A050144 := proc(n,k)
    if n < k then
        0;
    elif k =0 then
        if n =0 then
            0 ;
        else
            A000108(n-1) ;
        end if;
    elif k = 1 then
        add( procname(n-1-j,0)*A000108(j+1),j=0..n-1) ;
    elif k = 2 then
        add( procname(n-j,1)*A000108(j),j=0..n) ;
    else
        add( procname(n-1-j,k-1)*A000108(j),j=0..n-1) ;
    end if;
end proc:
seq(seq( A050144(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jul 30 2024

c[n_] := Binomial[2 n, n]/(n + 1);
t[n_, k_] := Which[k == 0, c[n - 1],
  k == 1, Sum[t[n - 1 - j, 0]*c[j + 1], {j, 0, n - 2}],
  k == 2, Sum[t[n - j, 1]*c[j], {j, 0, n - 1}],
  k > 2, Sum[t[n - 1 - j, k - 1] c[j + 1], {j, 0, n - 2}]]
t[0, 0] = 0;
Column[Table[t[n, k], {n, 0, 10}, {k, 0, n}]]
(* Clark Kimberling, Jul 30 2024 *)

For n > 0: Sum_{k>=0} T(n, k) = binomial(2*n-1, n); see A001700. - Philippe Deléham, Feb 13 2004 [Erroneous sum-formula deleted. R. J. Mathar, Jul 31 2024]
T(n, k)=0 if n < k; T(0, 0)=0, T(n, 0) = A000108(n-1) for n > 0; T(n, 1) = Sum_{j>=0} T(n-1-j, 0)*A000108(j+1); T(n, 2) = Sum_{j>=0} T(n-j, 1)*A000108(j); for k > 2, T(n, k) = Sum_{j>=0} T(n-1-j, k-1)*A000108(j+1). - Philippe Deléham, Feb 13 2004 [Corrected by Sean A. Irvine, Aug 08 2021]
For the column k=0, g.f.: x*C(x); for the column k=1, g.f.: x*C(x)*(C(x)-1); for the column k, k > 1, g.f.: x*C(x)^2*(C(x)-1)^(k-1); where C(x) = Sum_{n>=0} A000108(n)*x^n is g.f. for Catalan numbers, A000108. - Philippe Deléham, Feb 13 2004
T(n,0) = A033184(n,2). T(n,1) = A033184(n+1,3), T(n,k) = A033184(n+2,k+2) for k>=2. - R. J. Mathar, Jul 31 2024

A115147 Eighth convolution of A115140.

Original entry on oeis.org

1, -8, 20, -16, 2, 0, 0, 0, -1, -8, -44, -208, -910, -3808, -15504, -62016, -245157, -961400, -3749460, -14567280, -56448210, -218349120, -843621600, -3257112960, -12570420330, -48507033744, -187187399448, -722477682080, -2789279908316, -10772391370048

Offset: 0

Wolfdieter Lang, Jan 13 2006

Seiichi Manyama, Table of n, a(n) for n = 0..1670

Cf. A115139 - A115146, A115148, A115149.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-8*x+20*x^2-16*x^3+2*x^4 +(1-6*x+10*x^2-4*x^3)*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019

CoefficientList[Series[(1-8*x+20*x^2-16*x^3+2*x^4 +(1-6*x+10*x^2-4*x^3) *Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)

my(x='x+O('x^30)); Vec((1-8*x+20*x^2-16*x^3+2*x^4 +(1-6*x+10*x^2 -4*x^3)*sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019

((1-8*x+20*x^2-16*x^3+2*x^4 +(1-6*x+10*x^2-4*x^3)*sqrt(1-4*x))/2 ).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

O.g.f.: 1/c(x)^8 = P(9, x) - x*P(8, x)*c(x) with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(9, x)= 1-7*x+15*x^2-10*x^3+x^4 and P(8, x)=1-6*x+10*x^2-4*x^3.

a(n) = -C8(n-8), n>=8, with C8(n) = A003518(n+3) (eighth convolution of Catalan numbers). a(0)=1, a(1)=-8, a(2)=30, a(3)=-16, a(4)=2, a(5)=a(6)=a(7)=0. [1, -8, 20, -16, 2] is row n=8 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.

A099376 An inverse Chebyshev transform of x^3.

Original entry on oeis.org

0, 1, 4, 14, 48, 165, 572, 2002, 7072, 25194, 90440, 326876, 1188640, 4345965, 15967980, 58929450, 218349120, 811985790, 3029594040, 11338026180, 42550029600, 160094486370, 603784920024, 2282138106804, 8643460269248

Offset: 0

Paul Barry, Oct 13 2004

The sequence is 0,0,0,1,0,4,0,14,0,...with zeros restored. Second binomial transform of (-1)^n*A003518(n). Second binomial transform of expansion of x^3*c(-x)^8, where c(x) is g.f. of A000108. The g.f. is transformed to x^3 under the Chebyshev transformation A(x) -> (1/(1+x^2))*A(x/(1+x^2)). For a sequence b(n), this corresponds to taking Sum_{k=0..floor(n/2)} C(n-k,k) * (-1)^k * b(n-2k), or Sum_{k=0..n} C((n+k)/2,k) * b(k) * (-1)^((n-k)/2) * (1+(-1)^(n-k))/2.

Let X_n be the set of all noncrossing set partitions of an n-element set which either do not contain {n-1,n} as a block, or which do not contain the block {n} whenever 1 and n-1 are in the same block. For n>0, (-1)^n*a(n) gives the value of the Möbius function of X_{n+2} ordered by dual refinement between the discrete and the full partition. For example, X_3 is a chain consisting of 3 elements and its Möbius function between least and greatest element therefore takes the value a(1)=0. - Henri Mühle, Jan 10 2017

G. C. Greubel, Table of n, a(n) for n = 0..1000
H. Mühle, Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces, arXiv:1701.02109 [math.CO], 2017.

Cf. A000108, A000245, A003518, A071718.

Partial sums of A026016.

[Catalan(n+2) -2*Catalan(n+1): n in [0..30]]; // G. C. Greubel, May 05 2021

Table[CatalanNumber[n+2] -2CatalanNumber[n+1], {n, 0, 30}] (* or *)
Table[4 Binomial[2#+3, #]/(#+4) &[n-1], {n, 0, 30}] (* Michael De Vlieger, Jan 10 2017, latter after Harvey P. Dale at A002057 *)

{a(n)= if(n<1, 0, n++; 2* binomial(2*n, n-2)/n)} /* Michael Somos, Apr 11 2007 */

[catalan_number(n+2) -2*catalan_number(n+1) for n in (0..30)] # G. C. Greubel, May 05 2021

G.f.: (1-2*x)^4*( sqrt((1+2*x)/(1-2*x)) - 1)^8/(256*x^5).
a(n) = Sum_{k=0..n} (k+1)*C(n, (n-k)/2)*(-1)^k*( C(3, k) -3*C(2, k) +3*C(1, k) -C(0, k) )*(1+(-1)^(n-k))/(n+k+2).
a(n) = A002057(n-1). - Michael Somos, Jul 31 2005
Given an ellipse with eccentricity e and major and minor axis a and b respectively, then ((a-b)/ (a+b))^2 = 1*(e/2)^4 +4*(e/2)^6 +14*(e/2)^8 +48*(e/2)^10 + ... - Michael Somos, Apr 11 2007
E.g.f.: exp(2x)*(Bessel_I(1,2x) - Bessel_I(3,2x)). - Paul Barry, Jun 04 2007
D-finite with recurrence (n+3)*(n-1)*a(n) -2*n*(2*n+1)*a(n-1)=0. - R. J. Mathar, Sep 26 2012
a(n) = A000108(n+2) - 2*A000108(n+1) for n>0. - Henri Mühle, Jan 10 2017, corrected Sep 25 2021
G.f.: ( (1-2*x)*c(x) - (1-x) )/x^2, where c(x) is the gf of A000108. - G. C. Greubel, May 05 2021
From Peter Bala, Aug 30 2023: (Start)
a(n) = 2*n/((n+2)*(n+3)) * binomial(2*n+2, n+1).
a(n) = 2*Sum_{k = 0..n-1} 1/(n+1)*binomial(n+1, k)*binomial(n+1, k+2). (End)

cp's OEIS Frontend

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

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

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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.

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A130020 Triangle T(n,k), 0<=k<=n, read by rows given by [1,0,0,0,0,0,0,...] DELTA [0,1,1,1,1,1,1,...] where DELTA is the operator defined in A084938 .

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