A002057 -id:A002057 - cp's OEIS frontend

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 .

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

A006078 Number of triangulated (n+2)-gons rooted at an exterior edge.

Original entry on oeis.org

1, 1, 5, 12, 45, 143, 511, 1768, 6330, 22610, 81818, 297160, 1086813, 3991995, 14733435, 54587280, 203000094, 757398510, 2834519142, 10637507400, 40023665682, 150946230006, 570534682710, 2160865067312, 8199711750100

Offset: 2

N. J. A. Sloane, E. K. Lloyd (E.K.Lloyd(AT)soton.ac.uk)

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.

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
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.
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]
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]

G:=(4*(1-x-x^2)-(1-2*x)*(1-4*x)^(1/2)-3*(1-4*x^2)^(1/2))/8/x^2: Gser:=series(G,x=0,35): seq(coeff(Gser,x^n),n=2..28); # Emeric Deutsch, Dec 19 2004

g:=(4*(1-x-x^2)-(1-2*x)*(1-4*x)^(1/2)-3*(1-4*x^2)^(1/2))/8/x^2; gser := Series[g, {x, 0, 26}]; Drop[ CoefficientList[gser, x], 2] (* Jean-François Alcover, Apr 06 2012, after Emeric Deutsch *)
Drop[CoefficientList[Series[(4(1-x-x^2)- (1-2x)Sqrt[1-4x]- 3Sqrt[1- 4x^2])/(8x^2),{x,0,30}],x],2] (* Harvey P. Dale, Apr 07 2013 *)

Stockmeyer gives a g.f.
G.f.: (4*(1-x-x^2)-(1-2*x)(1-4*x)^(1/2)-3(1-4*x^2)^(1/2))/(8*x^2). - Emeric Deutsch, Dec 19 2004
a(n) ~ 2^(2*n-1) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 06 2014

More terms from Emeric Deutsch, Dec 19 2004

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

A052715 Expansion of e.g.f. (1-2x-sqrt(1-4x))/2 - x(1-2x-sqrt(1-4*x)) - x^2.

Original entry on oeis.org

0, 0, 0, 0, 24, 480, 10080, 241920, 6652800, 207567360, 7264857600, 282291609600, 12067966310400, 563171761152000, 28496491114291200, 1554354060779520000, 90929712555601920000, 5679609738088366080000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 671

Cf. A000108, A002057.

[n le 3 select 0 else 2*(n-3)*Factorial(n-1)*Catalan(n-2): n in [0..30]]; // G. C. Greubel, May 27 2022

spec := [S,{B=Union(Z,C),C=Prod(B,B),S=Prod(C,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[(1-2x-Sqrt[1-4x])/2-x(1-2x- Sqrt[ 1-4x])- x^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 18 2016 *)

[0]+[4*factorial(n-1)*binomial(2*n-5, n-4) for n in (1..30)] # G. C. Greubel, May 27 2022

D-finite with recurrence: a(1)=0; a(2)=0; a(3)=0; a(4)=24; 4*(-n-3+2*n^2)*a(n) +2*(-1-3*n)*a(n+1) +a(n+2) =0.

a(n) = n!*A002057(n-4). - R. J. Mathar, Oct 18 2013

A145596 Triangular array of generalized Narayana numbers: T(n, k) = 2binomial(n + 1, k + 1)binomial(n + 1, k - 1)/(n + 1).

Original entry on oeis.org

1, 2, 2, 3, 8, 3, 4, 20, 20, 4, 5, 40, 75, 40, 5, 6, 70, 210, 210, 70, 6, 7, 112, 490, 784, 490, 112, 7, 8, 168, 1008, 2352, 2352, 1008, 168, 8, 9, 240, 1890, 6048, 8820, 6048, 1890, 240, 9, 10, 330, 3300, 13860, 27720, 27720, 13860, 3300, 330, 10

Offset: 1

Peter Bala, Oct 14 2008

T(n,k) is the number of walks of n unit steps on the square lattice (i.e., each step in the direction either up (U), down (D), right (R) or left (L)) starting from (0,0) and finishing at points on the horizontal line y = 1, which remain in the upper half-plane y >= 0. An example is given in the Example section below.
The current array is the case r = 1 of the generalized Narayana numbers N_r(n, k) := (r + 1)/(n + 1)*binomial(n + 1, k + r)*binomial(n + 1, k - 1), which count walks of n steps from the origin to points on the horizontal line y = r that remain in the upper half-plane. Case r = 0 gives the table of Narayana numbers A001263 (but with row numbering starting at n = 0). For other cases see A145597 (r = 2), A145598 (r = 3) and A145599 (r = 4).
T(n,k) is the number of preimages of the permutation 21345...(n+2) under West's stack-sorting map that have exactly k descents. - Colin Defant, Sep 15 2018
T(n+k+1,k+1) equals the number of tilings of an octagon with internal angles of 135 degrees and sides of lengths n, k, 1, 1, n, k, 1, 1 using unit rhombi with internal angles 45 degrees and 135 degrees. See Elnitzky, Theorem 5.1. - Peter Bala, Apr 25 2022

			n\k|..1.....2....3.....4.....5.....6
====================================
.1.|..1
.2.|..2.....2
.3.|..3.....8....3
.4.|..4....20...20.....4
.5.|..5....40...75....40.....5
.6.|..6....70..210...210....70.....6
...
Row 3 entries:
T(3,1) = 3: the 3 walks from (0,0) to (-2,1) of three steps are LLU, LUL and ULL.
T(3,2) = 8: the 8 walks from (0,0) to (0,1) of three steps are UDU, UUD, ULR, URL, RLU, LRU, RUL and LUR.
T(3,3) = 3: the 3 walks from (0,0) to (2,1) of three steps are RRU, RUR and URR.
.
.
*......*......*......y......*......*......*
.
.
*......3......*......8......*......3......*
.
.
*......*......*......o......*......*......* x-axis
.
.

F. Cai, Q.-H. Hou, Y. Sun, and A. L. B. Yang, Combinatorial identities related to 2x2 submatrices of recursive matrices, arXiv:1808.05736 [math.CO],2018; Table 2.1 for k=1.
Colin Defant, Preimages under the stack-sorting algorithm, arXiv:1511.05681 [math.CO], 2015-2018; Graphs Combin., 33 (2017), 103-122.
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Serge Elnitsky, Rhombic tilings of polygons and classes of reduced words in Coxeter groups, J. Combin. Theory Ser. A, 77(2): 193-221, 1997.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
Toufik Mansour and Yidong Sun, Identities involving Narayana polynomials and Catalan numbers, arXiv:0805.1274 [math.CO], 2008.
Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 20.

Cf. A002057 (row sums), A001263, A108838, A145597, A145598, A145599, A145600.

/* As triangle */  [[2/(n+1)*Binomial(n+1,k+1)*Binomial(n+1,k-1): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Sep 15 2018

T:= (n, k) -> 2/(n+1)*binomial(n+1, k+1)*binomial(n+1, k-1):
for n from 1 to 10 do seq(T(n,k), k = 1..n) end do;

t[n_, k_]:=2/(n+1) Binomial[n+1, k+1] Binomial[n+1, k-1]; Table[t[n, k], {n, 3, 10}, {k, n}]//Flatten (* Vincenzo Librandi, Sep 15 2018 *)

T(n,k) = 2/(n + 1)*binomial(n + 1,k + 1)*binomial(n + 1,k - 1) for 1 <= k <= n. In the notation of [Guy], T(n,k) equals w_n(x,y) at (x,y) = (2*k - n - 1,1).
O.g.f. for column (k + 2): 2/(k + 1) * y^(k+2)/(1 - y)^(k+4) * Jacobi_P(k,2,1,(1 + y)/(1 - y)). The column generating functions begin: column 2: 2*y^2/(1 - y)^4; column 3: y^3*(3 + 2*y)/(1 - y)^6; column 4: y^4*(4 + 8*y + 2*y^2)/(1 - y)^8; the polynomials in the numerators are the row generating polynomials of array A108838.
O.g.f. for array: 1/(2*x*y^3) * (((1 + x)*y - 1)*sqrt(1 - 2*(1 + x)*y + (y - x*y)^2) + x^2*y^2 - 2*x*y + (1 - y)^2) = x*y + (2*x + 2*x^2)*y^2 + (3*x + 8*x^2 + 3*x^3)*y^3 + (4*x + 20*x^2 + 20*x^3 + 4*x^4)*y^4 + ... .
Row sums A002057.
Identities for row polynomials R_n(x) = Sum_{k = 1..n} T(n,k)*x^k (compare with the results in section 1 of [Mansour & Sun]):
x*R_(n-1)(x) = 2*(n - 1)/((n + 1)*(n + 2)) * Sum_{k = 0..n} binomial(n + 2,k) * binomial(2*n - k,n) * (x - 1)^k;
R_n(x) = Sum_{k = 0..floor((n-1)/2)} binomial(n, 2*k + 1) * Catalan(k + 1) * x^(k+1)*(1 + x)^(n-2k-1);
Sum_{k = 1..n} (-1)^(n-k)*binomial(n,k)*R_k(x)*(1 + x)^(n-k) = x^m*Catalan(m) if n = 2*m - 1 is odd, otherwise the sum is zero.
Sum_{k = 1..n} (-1)^(k+1)*binomial(n,k)*R_k(x^2)*(1 + x)^(2*(n-k)) = R_n(1)*x^(n+1) = 4/(n + 3)*binomial(2*n + 1,n - 1)*x^(n+1) = A002057(n-1)*x^(n+1).
Row generating polynomial R_(n+1)(x) = 2/(n + 2)*x*(1 - x)^n * Jacobi_P(n,2,2,(1 + x)/(1 - x)). - Peter Bala, Oct 31 2008
G.f. satisfies x^3*y*A(x,y)^2-A(x,y)*(x^2*y^2+(-2)*x*y+x^2+(-2)*x+1)+x = 0. - Vladimir Kruchinin, Oct 11 2020
The array can be extended to negative values of n: T(-n,k) = 2*binomial(-n + 1, k + 1)*binomial(-n + 1, k - 1)/(-n + 1) = -A108838(n+k-1,k-1) for n >= 2. - Peter Bala, Apr 26 2022
From Sergii Voloshyn, Dec 18 2024: (Start)
Let E be the operator x^2*D*(1/x)*D, where D denotes the derivative operator d/dx. Then 48(1+n)/((n+2)!(n+4)!)* E^n(x^2/(1 - x)^5) = (row n generating polynomial)/(1 - x)^(2*n+5) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1).
For example, when n = 3 we have 1/3150*E^3(x^2/(1 - x)^5) = x^2 (4 + 20x + 20x^2 + 4x^3)/(1 - x)^11. (End)

A052721 Expansion of e.g.f. x(1-2x)(1 - 2x - sqrt(1-4*x))/2 - x^3.

Original entry on oeis.org

0, 0, 0, 0, 0, 120, 2880, 70560, 1935360, 59875200, 2075673600, 79913433600, 3387499315200, 156883562035200, 7884404656128000, 427447366714368000, 24869664972472320000, 1545805113445232640000, 102232975285590589440000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 677

Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052722, A052723.

Cf. A000108, A002057.

spec := [S,{C=Union(B,Z),B=Prod(C,C),S=Prod(B,B,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Table[If[n<5, 0, 2*n*(n-2)!*(n-4)*CatalanNumber[n-3]], {n,0,30}] (* G. C. Greubel, May 28 2022 *)

def A052721(n):
    if (n<5): return 0
    else: return 2*n*factorial(n-2)*(n-4)*catalan_number(n-3)
[A052721(n) for n in (0..30)] # G. C. Greubel, May 28 2022

D-finite with recurrence: a(1)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=120, a(6)=2880, (n+2)*a(n+2) = (6*n^2 + 8*n - 8)*a(n+1) + (40 + 44*n = 4*n^2 - 8*n^3)*a(n).
a(n) = 2*Pi^(-1/2)*4^(n-3)*Gamma(n-5/2)*n*(n-4) for n>3.  - Mark van Hoeij, Oct 30 2011
a(n) = n!*A002057(n-5). - R. J. Mathar, Oct 18 2013
From G. C. Greubel, May 28 2022: (Start)
G.f.: 4!*x*(d/dx)( x^5 * Hypergeometric2F0([2, 5/2], [], 4*x) ).
E.g.f.: (x/2)*(1 - 4*x + 2*x^2 - (1-2*x)*sqrt(1-4*x)). (End)

A071721 Expansion of (1+x^2C^2)C^2, where C = (1-(1-4x)^(1/2))/(2x) is g.f. for Catalan numbers, A000108.

Original entry on oeis.org

1, 2, 6, 18, 56, 180, 594, 2002, 6864, 23868, 83980, 298452, 1069776, 3863080, 14040810, 51325650, 188574240, 695987820, 2579248980, 9593714460, 35804293200, 134032593240, 503154100020, 1893689067348, 7144084508256

Offset: 0

N. J. A. Sloane, Jun 06 2002

nonn

a(n) = A138156(n) - 4*A138156(n-1). - Alzhekeyev Ascar M, Jul 19 2011
Apparently, for n>=1, the sum of the heights of the first and last peaks in all Dyck n-paths (in paths with one peak the height counts as both first and last). - David Scambler, Oct 05 2012
For n>=1, a(n) is the total number of nonempty subtrees over all binary trees having n+1 internal nodes.  Here, a binary tree is a full (each node has two or zero children), rooted, plane (ordered), unlabeled tree.  An empty subtree is a tree attached to the root that consists only of an external node.  a(n) = 2*A002057(n-2) + A068875(n). - Geoffrey Critzer, Sep 16 2013
From Colin Defant, Sep 15 2018: (Start)
a(n) is the number of permutations pi of [n+1] such that s(pi) avoids the patterns 132, 231, 312, and 321, where s denotes West's stack-sorting map.
a(n) is the number of permutations on [n+1] that avoid the patterns 1342, 2341, 3142, 3241, 3412, and 3421. (End)

			G.f. = 1 + 2*x + 6*x^2 + 18*x^3 + 56*x^4 + 180*x^5 + 594*x^6 + 2002*x^7 + ... - _Michael Somos_, Apr 22 2022

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Stoyan Dimitrov, On permutation patterns with constrained gap sizes, arXiv:2002.12322 [math.CO], 2020.
Ling Gao, Graph assembly for spider and tadpole graphs, Master's Thesis, Cal. State Poly. Univ. (2023). See p. 32.
Boram Park and Seonjeong Park, Shellable posets arising from even subgraphs of a graph, arXiv preprint arXiv:1705.06423 [math.CO], 2017.
Seonjeong Park, Real toric manifolds and shellable posets arising from graphs, 2018.

Cf. A000108, A000245, A001622, A002421, A068875.
Row sums of triangles A319251, A319252.
gf=(1+x^2*C^2)*C^m: A000782 (m=1), this sequence (m=2), A071722 (m=3), A071723 (m=4).

a := n -> `if`(n=0, 1, 6*binomial(2*n, n-1)/(n+2));
seq(a(n), n=0..24); # Peter Luschny, Jun 28 2018

Join[{1},Table[6n CatalanNumber[n]/(n+2),{n,30}]] (* Harvey P. Dale, Jun 05 2012 *)
nn=20;t=(1-(1-4x)^(1/2))/(2x);CoefficientList[Series[D[1+x (y t -y+1)^2,y]/.y->1,{x,0,nn}],x] (* Geoffrey Critzer, Sep 16 2013 *)

{a(n) = if(n<1, n==0, 6*n*(2*n)!/(n!*(n + 1)!*(n + 2)))}; /* Michael Somos, Apr 22 2022 */

a = lambda n: n*(n+1)*hypergeometric([1-n, 2-n], [4], 1) if n>0 else 1
[simplify(a(n)) for n in range(25)] # Peter Luschny, Nov 19 2014

a(n) = 6n * (2n)! / [(n+2)n!(n+1)! ], n>0. In terms of Catalan numbers (A000108), a(n) = 6n*Cat(n)/(n+2), n>0. - Ralf Stephan, Mar 11 2004
a(n) = n*(n+1)*hypergeom([1-n, 2-n], [4], 1) for n>=1. - Peter Luschny, Nov 19 2014
D-finite with recurrence -(n+2)*(n-1)*a(n) +2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Jul 18 2017
a(n) = 2*Cat(n+1) - 2*Cat(n) = 2*A000245(n) for n>=1. - Colin Defant, Jun 27 2018
From Amiram Eldar, Mar 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 23/18 + 7*Pi/(27*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 43/50 - 82*sqrt(5)*log(phi)/375, where phi is the golden ratio (A001622). (End)
From Michael Somos, Apr 22 2022: (Start)
G.f.: (1 - 3*x + x^2 - (1 - x) * sqrt(1 - 4*x))/x^2.
G.f.: (2 - 2*x + x^2)/(1 - 3*x + x^2 + (1 - x)*sqrt(1 - 4*x)).
G.f.: 1 + 1/((1 - x)/(1 - sqrt(1 - 4*x)) - 1/2).
a(n) = b(n+1) - b(n) for all n in Z if b(0) = 2, b(-1) = -1, a(0) = 0, a(-1) = 3, a(-2) = -1 where b = A068875.
0 = a(n)*(+16*a(n+1) -58*a(n+2) +18*a(n+3)) +a(n+1)*(+18*a(n+1) +15*a(n+2) -13*a(n+3)) +a(n+2)*(+3*a(n+2) +a(n+3)) for all n in Z if a(0) = 0, a(-1) = 3, a(-2) = -1. (End)

A114277 Sum of the lengths of the second ascents in all Dyck paths of semilength n+2.

Original entry on oeis.org

1, 5, 19, 67, 232, 804, 2806, 9878, 35072, 125512, 452388, 1641028, 5986993, 21954973, 80884423, 299233543, 1111219333, 4140813373, 15478839553, 58028869153, 218123355523, 821908275547, 3104046382351, 11747506651599

Offset: 0

Emeric Deutsch, Nov 20 2005

nonn

Also number of Dyck paths of semilength n+4 having length of second ascent equal to three. Example: a(1)=5 because we have UD(UUU)DUDDD, UD(UUU)DDUDD, UD(UUU)DDDUD, UUD(UUU)DDDD and UUDD(UUU)DDD (second ascents shown between parentheses). Partial sums of A002057. Column 3 of A114276. a(n)=absolute value of A104496(n+3).

Also number of Dyck paths of semilength n+3 that do not start with a pyramid (a pyramid in a Dyck path is a factor of the form U^j D^j (j>0), starting at the x-axis; here U=(1,1) and D=(1,-1); this definition differs from the one in A091866). Equivalently, a(n)=A127156(n+3,0). Example: a(1)=5 because we have UUDUDDUD, UUDUDUDD, UUUDUDDD, UUDUUDDD and UUUDDUDD. - Emeric Deutsch, Feb 27 2007

			a(3)=5 because the total length of the second ascents in UD(U)DUD, UD(UU)DD, UUDD(U)D, UUD(U)DD and UUUDDD (shown between parentheses) is 5.

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

Cf. A002057, A114276, A104496, A127156, A279557.

Cf. A014137 (n=1), A014138 (n=2), A001453 (n=3), this sequence (n=4), A143955 (n=5), A323224 (array).

a:=n->4*sum(binomial(2*j+3,j)/(j+4),j=0..n): seq(a(n),n=0..28);

Table[4*Sum[Binomial[2j+3,j]/(j+4),{j,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 19 2012 *)

from functools import cache
@cache
def B(n, k):
    if n <= 0 or k <= 0: return 0
    if n == k: return 1
    return B(n - 1, k) + B(n, k - 1)
def A114277(n): return B(n + 5, n + 1)
print([A114277(n) for n in range(24)]) # Peter Luschny, May 16 2022

a(n) = 4*Sum_{j=0..n} binomial(2*j+3, j)/(j+4).
G.f.: C^4/(1-z), where C=(1-sqrt(1-4*z))/(2*z) is the Catalan function.
a(n) = c(n+3) - (c(0) + c(1) + ... + c(n+2)), where c(k)=binomial(2k,k)/(k+1) is a Catalan number (A000108). - Emeric Deutsch, Feb 27 2007
D-finite with recurrence: n*(n+4)*a(n) = (5*n^2 + 14*n + 6)*a(n-1) - 2*(n+1)*(2*n+3)*a(n-2). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 2^(2*n+7)/(3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012
a(n) = exp((2*i*Pi)/3)-4*binomial(2*n+5,n+1)*hypergeom([1,3+n,n+7/2],[n+2,n+6],4)/ (n+5). - Peter Luschny, Feb 26 2017
a(n-1) = Sum_{i+j+k+lA000108 Catalan number. - Yuchun Ji, Jan 10 2019

More terms from Emeric Deutsch, Feb 27 2007

A000150 Number of dissections of an n-gon, rooted at an exterior edge, asymmetric with respect to that edge.

Original entry on oeis.org

0, 0, 1, 2, 7, 20, 66, 212, 715, 2424, 8398, 29372, 104006, 371384, 1337220, 4847208, 17678835, 64821680, 238819350, 883629164, 3282060210, 12233125112, 45741281820, 171529777432, 644952073662, 2430973096720, 9183676536076

Offset: 0

N. J. A. Sloane

Number of Dyck paths of length 2n having an odd number of peaks at even height. Example: a(3)=2 because we have UDU(UD)D and U(UD)DUD, where U=(1,1), D=(1,-1) and the peaks at even height are shown between parentheses. - Emeric Deutsch, Nov 13 2004
For n>=1, a(n) is the number of unordered binary trees with n internal nodes in which the left subtree is distinct from the right subtree. - Geoffrey Critzer, Feb 21 2013
Assuming offset -1 this is an analog of A275166: pairs of distinct Catalan numbers with index sum n. - R. J. Mathar, Jul 19 2016

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
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. 78, (3.5.26).
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=0..200
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.
Krishna Menon and Anurag Singh, Grassmannian permutations avoiding identity, arXiv:2212.13794 [math.CO], 2022.
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]
Index entries for sequences related to Lyndon words

a(n) = T(2n+2, n), array T as in A051168, a count of Lyndon words.
Cf. A051168, A005430.
Cf. A007595.
A diagonal of the square array described in A051168.

nn=20;CoefficientList[Series[x/2(((1-(1-4x)^(1/2))/(2x))^2-(1-(1-4x^2)^(1/2))/(2x^2)),{x,0,nn}],x]  (* Geoffrey Critzer, Feb 21 2013 *)

Let c(x) = (1-sqrt(1-4*x))/(2*x) = g.f. for Catalan numbers (A000108), let d(x) = 1+x*c(x^2). Then g.f. is (c(x)-d(x))/2.
G.f.: (sqrt(1-4*z^2) - sqrt(1-4*z) - 2*z)/(4*z). - Emeric Deutsch, Nov 13 2004
With c(x) defined as above: g.f. = x*(c(x)^2/2 - c(x^2)/2). - Geoffrey Critzer, Feb 21 2013
a(n) = ( 2^(n-3)/sqrt(Pi) ) * ( 4*2^n*GAMMA(n+1/2)/GAMMA(n+2) + ((-1)^n - 1)*GAMMA(n/2)/GAMMA(n/2 + 3/2) ) for n>0. - Mark van Hoeij, Nov 11 2009
a(n) ~ 2^(2*n-1) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 10 2014
a(2n) = A000108(2n) / 2; a(2n+1) = ( A000108(2n+1) - A000108(n) ) / 2. - John Bodeen, Jun 24 2015
D-finite with recurrence +n*(n+1)*(n-2)^2*a(n) -2*n*(2*n-5)*(n-1)^2*a(n-1) -4*n*(n-2)^3*a(n-2) +8*(2*n-5)*(n-3)*(n-1)^2*a(n-3)=0. - R. J. Mathar, Oct 28 2021

Additional comments from Clark Kimberling

cp's OEIS Frontend

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

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A006078 Number of triangulated (n+2)-gons rooted at an exterior edge.

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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.

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A052715 Expansion of e.g.f. (1-2*x-sqrt(1-4*x))/2 - x*(1-2*x-sqrt(1-4*x)) - x^2.

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A145596 Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1).

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A052721 Expansion of e.g.f. x*(1-2*x)*(1 - 2*x - sqrt(1-4*x))/2 - x^3.

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A052715 Expansion of e.g.f. (1-2x-sqrt(1-4x))/2 - x(1-2x-sqrt(1-4*x)) - x^2.

A145596 Triangular array of generalized Narayana numbers: T(n, k) = 2binomial(n + 1, k + 1)binomial(n + 1, k - 1)/(n + 1).

A052721 Expansion of e.g.f. x(1-2x)(1 - 2x - sqrt(1-4*x))/2 - x^3.

A071721 Expansion of (1+x^2C^2)C^2, where C = (1-(1-4x)^(1/2))/(2x) is g.f. for Catalan numbers, A000108.