A000108 -id:A000108 - cp's OEIS frontend

A131428 a(n) = 2*C(n) - 1, where C(n) = A000108(n) are the Catalan numbers.

1, 1, 3, 9, 27, 83, 263, 857, 2859, 9723, 33591, 117571, 416023, 1485799, 5348879, 19389689, 70715339, 259289579, 955277399, 3534526379, 13128240839, 48932534039, 182965127279, 686119227299, 2579808294647, 9723892802903, 36734706144303, 139067101832007

Offset: 0

Gary W. Adamson, Jul 10 2007

nonn

Starting (1, 3, 9, 27, 83, ...), = row sums of triangle A136522. - Gary W. Adamson, Jan 02 2008
Hankel transform is A171552. - Paul Barry, Dec 11 2009
Apparently, for n >= 1, the maximum peak height minus the maximum valley height summed over all Dyck n-paths (with max valley height deemed zero if no valleys). - David Scambler, Oct 05 2012
Apparently for n > 1 the number of fixed points in all Dyck (n-1)-paths. A fixed point occurs when a vertex of a Dyck k-path is also a vertex of the path U^kD^k. - David Scambler, May 01 2013

			a(3) = 9 = 2*C(3) - 1 = 2*5 - 1, where C refers to the Catalan numbers, A000108.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000108, A131427, A131429.
Cf. A131428, A118976, A136522.
Cf. A080233, A156644.

List([0..25], n-> 2*Binomial(2*n,n)/(n+1) - 1); # G. C. Greubel, Aug 12 2019

[2*Catalan(n) -1: n in [0..25]]; // G. C. Greubel, Aug 12 2019

seq(2*binomial(2*n,n)/(n+1)-1, n=0..25); # Emeric Deutsch, Jul 25 2007

2CatalanNumber[Range[0,25]]-1  (* Harvey P. Dale, Apr 17 2011 *)

vector(25, n, n--; 2*binomial(2*n,n)/(n+1) - 1) \\ G. C. Greubel, Aug 12 2019

[2*catalan_number(n) -1 for n in (0..25)] # G. C. Greubel, Aug 12 2019

Right border of triangle A131429.
From Emeric Deutsch, Jul 25 2007: (Start)
a(n) = 2*binomial(2*n,n)/(n+1) - 1.
G.f.: (1-sqrt(1-4*x))/x - 1/(1-x). (End)
(1, 3, 9, 27, 83, ...) = row sums of A118976. - Gary W. Adamson, Aug 31 2007
Row sums of triangle A131428 starting (1, 3, 9, 27, 83, ...). - Gary W. Adamson, Aug 31 2007
Starting with offset 1 = Narayana transform (A001263) of [1,2,2,2,...]. - Gary W. Adamson, Jul 29 2011
D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +3*(-n+1)=0. - R. J. Mathar, Nov 22 2024
a(n) = Sum_{k=0..n} ( binomial(n,k) - binomial(n,k-1) )^2 = Sum_{k=0..n} A080233(n,k)^2 = Sum_{k=0..n} A156644(n,k)^2. - Seiichi Manyama, Mar 25 2025

More terms from Emeric Deutsch, Jul 25 2007

A076025 Expansion of g.f.: (1-3xC)/(1-4xC) where C = (1 - sqrt(1-4x))/(2x) = g.f. for Catalan numbers A000108.

Original entry on oeis.org

1, 1, 5, 26, 137, 726, 3858, 20532, 109361, 582782, 3106550, 16562668, 88314634, 470942044, 2511443268, 13393472616, 71428622337, 380940866574, 2031641406798, 10835261623356, 57787472903502, 308197667445204, 1643712737618748, 8766437439778776, 46754218658948922

Offset: 0

N. J. A. Sloane, Oct 29 2002

From Paul Barry, Sep 23 2009: (Start)
The Hankel transform of this sequence is 3n+1 or 1,4,7,10,... (A016777).
The Hankel transform of the aeration of this sequence is A016777 doubled, that is, 1,1,4,4,7,7,...
In general, the Hankel transform of [x^n](1-r*xc(x))/(1-(r+1)*xc(x)) is rn+1, and that of the corresponding aerated sequence is the doubled sequence of rn+1. (End)

L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.

G. C. Greubel, Table of n, a(n) for n = 0..1000
José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, On One-Parameter Catalan Arrays, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.1.
Richard Ehrenborg, Gábor Hetyei, and Margaret Readdy, Catalan-Spitzer permutations, arXiv:2310.06288 [math.CO], 2023. See p. 20.

Cf. A000108, A001700, A049027, A076026.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1- 3*Sqrt(1-4*x))/(2-4*Sqrt(1-4*x)) )); // G. C. Greubel, May 04 2019

CoefficientList[Series[(1-3*Sqrt[1-4*x])/(2-4*Sqrt[1-4*x]),{x,0,30}],x] (* Vaclav Kotesovec, Dec 09 2013 *)
Flatten[{1,Table[FullSimplify[(2*n)! * Hypergeometric2F1Regularized[1, n+1/2, n+2, 3/4] / (16*n!) + 2^(4*n-1)/3^(n+1)], {n,1,30}]}] (* Vaclav Kotesovec, Dec 09 2013 *)

my(x='x+O('x^30)); Vec((1-3*sqrt(1-4*x))/(2-4*sqrt(1-4*x))) \\ G. C. Greubel, May 04 2019

((1-3*sqrt(1-4*x))/(2-4*sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 04 2019

a(n+1) = Sum_{k=0..n} 3^k*binomial(2n+1, n-k)*2*(k+1)/(n+k+2). - Paul Barry, Jun 22 2004
a(n+1) = Sum_{k=0..n} A039598(n,k)*3^k. - Philippe Deléham, Mar 21 2007
a(n) = Sum_{k=0..n} A039599(n,k)*A015518(k), for n >= 1. - Philippe Deléham, Nov 22 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>=1, a(n+1)=(-1)^n*charpoly(A,-4). - Milan Janjic, Jul 08 2010
From Gary W. Adamson, Jul 25 2011: (Start)
a(n) = upper left term in M^(n-1), M = an infinite square production matrix as follows:
  5, 1, 0, 0, 0, ...
  1, 1, 1, 0, 0, ...
  1, 1, 1, 1, 0, ...
  1, 1, 1, 1, 1, ...
  ... (End)
D-finite with recurrence: 3*n*a(n) +2*(9-14*n)*a(n-1) +32*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 14 2011
a(n) ~ 2^(4*n-1)/3^(n+1). - Vaclav Kotesovec, Dec 09 2013
The sequence is the INVERT transform of A049027: (1, 4, 17, 74, 326, ...) and the third INVERT transform of the Catalan sequence (1, 2, 5, ...). - Gary W. Adamson, Jun 23 2015
O.g.f.: A(x) = (1 - 1/2*Sum_{n >= 1} binomial(2*n,n)*x^n)/(1 - Sum_{n >= 1} binomial(2*n,n)*x^n). - Peter Bala, Sep 01 2016

A076035 G.f.: 1/(1-4xC) where C = (1/2-1/2(1-4x)^(1/2))/x = g.f. for Catalan numbers A000108.

Original entry on oeis.org

1, 4, 20, 104, 548, 2904, 15432, 82128, 437444, 2331128, 12426200, 66250672, 353258536, 1883768176, 10045773072, 53573890464, 285714489348, 1523763466296, 8126565627192, 43341046493424, 231149891614008, 1232790669780816, 6574850950474992, 35065749759115104

Offset: 0

N. J. A. Sloane, Oct 29 2002

nonn

The Hankel transform of this sequence and that of the aerated sequence with g.f. 1/(1-4x^2*c(x^2)) is 4^n. In general, the expansions of 1/(1-k*x*c(x)) and 1/(1-k*x^2*c(x^2)) have Hankel transform k^n. - Paul Barry, Jan 20 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200
S. B. Ekhad, M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).

Cf. A000108, A000984, A007854, A076036.

CatalanNumber := n -> binomial(2*n,n)/(n+1):
h := (n, m) -> hypergeom([1+m, m-n], [m+n+2], -3):
a := n -> CatalanNumber(n)*(h(n,0) + 6*n/(n+2)*h(n,1)):
seq(simplify(a(n)), n=0..23); # Peter Luschny, Dec 09 2018

CoefficientList[Series[1/(1-4*x*(1-Sqrt[1-4*x])/(2*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

a(n) = sum{k=0..n, 3^k*C(2n, n-k)(2k+1)/(n+k+1)}. - Paul Barry, Jun 22 2004
a(n) = Sum_{k, 0<=k<=n} A106566(n, k)*4^k. - Philippe Deléham, Sep 01 2005
a(n) = if(n=0,1,sum{k=1..n, C(2n-k-1,n-k)*k*4^k/n}). - Paul Barry, Jan 20 2007
a(n) = Sum{k, 0<=k<=n}A039599(n,k)*3^k. - Philippe Deléham, Sep 08 2007
a(0)=1, a(n)=(16*a(n-1)-4*A000108(n-1))/3. - Philippe Deléham, Nov 27 2007
3*n*a(n) +2*(9-14*n)*a(n-1) +32*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011 [proved by Ekhad & Yang, see link]
a(n) ~ 2^(4*n+1) / 3^(n+1). - Vaclav Kotesovec, Feb 13 2014
Conjecture: a(n) = 4*A076025(n), n>0. - R. J. Mathar, Apr 01 2022

A129442 Expansion of c(x)c(xc(x)) where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 6, 21, 80, 322, 1348, 5814, 25674, 115566, 528528, 2449746, 11485068, 54377288, 259663576, 1249249981, 6049846848, 29469261934, 144293491564, 709806846980, 3506278661820, 17385618278700, 86500622296800

Offset: 0

Philippe Deléham, May 28 2007, Jun 20 2007

nonn

The sequence b(n) = [0,1,2,6,21,80,322,1348,...] for n >= 0 is the Catalan transform of Catalan numbers C(n-1), with C(-1)=0; Sum_{k=0..n} A106566(n,k) * A000108(k-1) = b(n).
A121988 is an essentially identical sequence. - R. J. Mathar, Jun 13 2008
Catalan transform of A014137. - R. J. Mathar, Nov 11 2008

			G.f. = 1 + 2*x + 6*x^2 + 21*x^3 + 80*x^4 + 322*x^5 + 1349*x^6 + ... - _Michael Somos_, May 28 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Cf. A000108, A014137, A106566, A121988, A236339.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(2*Sqrt(1-4*x)-1))/(2*x) )); // G. C. Greubel, Feb 06 2024

c := proc (x) options operator, arrow; (1/2)*(1-sqrt(1-4*x))/x end proc; G := simplify(c(x)*c(x*c(x))); Gser := series(G, x = 0, 28); seq(coeff(Gser, x, n), n = 0 .. 24) # Emeric Deutsch, Jun 20 2007

a[n_]:= Sum[ Binomial[2n -k-1, n-1]*Binomial[2k-2, k-1], {k, n}]/n;
Array[a, 23] (* Robert G. Wilson v, Jul 18 2007 *)

def A129442_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-sqrt(2*sqrt(1-4*x)-1))/(2*x) ).list()
A129442_list(40) # G. C. Greubel, Feb 06 2024

a(n-1) = (1/n)*Sum_{k=1..n} binomial(2*n-k-1, n-1)*binomial(2*k-2, k-1).
G.f.: (1-sqrt(2*sqrt(1-4*x)-1))/(2*x). - Emeric Deutsch, Jun 20 2007 Corrected by Stefan Forcey (sforcey(AT)tnstate.edu), Aug 02 2007
From Vaclav Kotesovec, Oct 20 2012: (Start)
Recurrence: 3*n*(n+1)*a(n) = 14*n*(2*n-1)*a(n-1) - 4*(4*n-5)*(4*n-3)*a(n-2).
a(n) ~ 2^(4*n+3/2)/(3^(n+1/2)*sqrt(Pi)*n^(3/2)). (End)
0 = +a(n)*(+a(n+1)*(+262144*a(n+2) -275968*a(n+3) +52608*a(n+4)) +a(n+2)*(-50176*a(n+2) +107680*a(n+3) -27930*a(n+4)) +a(n+3)*(-6006*a(n+3) +2574*a(n+4))) +a(n+1)*(+a(n+1)*(-17920*a(n+2) +21952*a(n+3) -4494*a(n+4)) +a(n+2)*(+5152*a(n+2) -15820*a(n+3) +4611*a(n+4)) +a(n+3)*(+1470*a(n+3) -630*a(n+4))) +a(n+2)*(+a(n+2)*(+42*a(n+2) +129*a(n+3) -63*a(n+4)) +a(n+3)*(-63*a(n+3) +27*a(n+4))) for n>=0. - Michael Somos, May 28 2023
From Seiichi Manyama, Jan 10 2023: (Start)
G.f.: (1/x) * Series_Reversion( x * (1-x) * (1-x+x^2) ).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(n+k,k) * binomial(3*n-k+1,n-2*k). (End)

More terms from Emeric Deutsch, Jun 20 2007

A187357 Catalan trisection: A000108(3*n), n >= 0.

Original entry on oeis.org

1, 5, 132, 4862, 208012, 9694845, 477638700, 24466267020, 1289904147324, 69533550916004, 3814986502092304, 212336130412243110, 11959798385860453492, 680425371729975800390, 39044429911904443959240, 2257117854077248073253720, 131327898242169365477991900, 7684785670514316385230816156, 451959718027953471447609509424

Offset: 0

Wolfdieter Lang, Mar 09 2011

Trisection of a sequence, given by its real o.g.f. G(x), is accomplished by
G(x) = G0(x^3) + x*G1(x^3) + (x^2)*G2(x^3), with the following solutions (using r := exp(2*Pi*i/3) = (-1 + sqrt(3)*i)/2):
G0(x) = (G(x^(1/3) + (G(r*x^(1/3)) + c.c.))/3,
G1(x) = (G(x^(1/3)) + ((1/r)*G(r*x^(1/3)) + c.c.))/(3*x^(1/3)),
G2(x) = (G(x^(1/3)) + (r*G(r*x^(1/3)) + c.c.))/(3*x^(2/3)),
where c.c. denotes the complex conjugate of the preceding expression.
See also the J. Arndt link, sect. 36.1.4,p.688: "Multisection by selecting terms with exponents s mod M", with M=3, where the o.g.f.s for the M-sected sequences with interspersed zeros are given for the general case.

Joerg Arndt, Fxtbook.

Cf. A000108, A024492, A048990, A187358 (C(3*n+1)), A187359 (C(3*n+2)/2), A208745.

Table[CatalanNumber[3*n], {n, 0, 20}] (* Amiram Eldar, Mar 16 2022 *)

a(n) = C(3*n), n >= 0, with C(n):= A000108(n) (Catalan).
O.g.f.: G0(x) = (sqrt(2*sqrt(1 + 4*x^(1/3) + 16*x^(2/3)) - (1 - 4*x^(1/3))) - sqrt(1 - 4*x^(1/3)))/(6*x^(1/3)).
From Ilya Gutkovskiy, Jan 13 2017: (Start)
E.g.f.: 3F3(1/6,1/2,5/6; 2/3,1,4/3; 64*x).
a(n) ~ 64^n/(3*sqrt(3*Pi)*n^(3/2)). (End)
D-finite with recurrence n*(3*n-1)*(3*n+1)*a(n) -8*(6*n-5)*(6*n-1)*(2*n-1)*a(n-1)=0. - R. J. Mathar, Feb 21 2020
Sum_{n>=0} a(n)/4^n = (4/3)^(3/4) (A208745). - Amiram Eldar, Mar 16 2022
a(n) = Product_{1 <= i <= j <= 3*n-1} (3*i + j + 2)/(3*i + j - 1). - Peter Bala, Feb 22 2023

A025225 a(n) = a(1)a(n-1) + a(2)a(n-2) + ...+ a(n-1)a(1) for n >= 2. Also a(n) = (2^n)C(n-1), where C = A000108 (Catalan numbers).

Original entry on oeis.org

2, 4, 16, 80, 448, 2688, 16896, 109824, 732160, 4978688, 34398208, 240787456, 1704034304, 12171673600, 87636049920, 635361361920, 4634400522240, 33985603829760, 250420238745600, 1853109766717440, 13765958267043840, 102618961627054080, 767411365211013120

Offset: 1

Clark Kimberling

Number of generators of degree n of the Hopf algebra of 2-colored planar binary trees. Also, dimensions of the graded components of the primitive Lie algebra of the same Hopf algebra. - Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008

Alois P. Heinz, Table of n, a(n) for n = 1..350
Suzanne Bobzien, The combinatorics of the Stoic conjunction: Hipparchus refuted and Chrysippus vindicated, Oxford Studies in Ancient Philosophy, Vol. XL, Summer 2011, pp. 157-188.
L. Guo and W. Y. Sit, Enumeration and generating functions of Rota-Baxter Words, Math. Comput. Sci. 4 (2010) 313-337, remark 3.3.
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 653.
J.-C. Novelli and J.-Y. Thibon, Free quasi-symmetric functions and descent algebras for wreath products and noncommutative multi-symmetric functions, arXiv:0806.3682 [math.CO], 2008.
Volkan Yildiz, Counting false entries in truth tables of bracketed formulas connected by m-implication, arXiv:1203.4645 [math.CO], 2012.
Volkan Yildiz, General combinatorical structure of truth tables of bracketed formulas connected by implication, arXiv:1205.5595 [math.CO], 2012.

Essentially identical to A115125.

Cf. A052701.

[2^n*Catalan(n-1): n in [1..30]]; // Vincenzo Librandi, Nov 06 2016

a:= n-> (2^n)*binomial(2*n-2, n-1)/n:
seq(a(n), n=1..25); # Alois P. Heinz, Jan 27 2012

InverseSeries[Series[y/2-y^2/2, {y, 0, 24}], x] (* then A(x)=y(x) *) (* Len Smiley, Apr 13 2000 *)
a[n_] := 2^n*CatalanNumber[n - 1]; Table[a[n], {n, 1, 23}] (* Jean-François Alcover, Jul 09 2013 *)

a(n)=polcoeff((1-sqrt(1-8*x+x*O(x^n)))/2,n)

G.f.: (1-sqrt(1-8*x))/2. - Michael Somos, Jun 08 2000
Given g.f. C(x) and given A(x)= g.f. of A100238, then B(x)=A(x)-1-x satisfies B(x)=x-C(x*B(x)). - Michael Somos, Sep 07 2005
n*a(n) + 4*(-2*n+3)*a(n-1) = 0. - R. J. Mathar, Feb 25 2015
a(n) = 2*A052701(n). - Alois P. Heinz, Feb 16 2025

Typo in definition corrected by R. J. Mathar, Aug 11 2008

A115140 O.g.f. inverse of Catalan A000108 o.g.f.

Original entry on oeis.org

1, -1, -1, -2, -5, -14, -42, -132, -429, -1430, -4862, -16796, -58786, -208012, -742900, -2674440, -9694845, -35357670, -129644790, -477638700, -1767263190, -6564120420, -24466267020, -91482563640, -343059613650, -1289904147324, -4861946401452, -18367353072152

Offset: 0

Wolfdieter Lang, Jan 13 2006

Seiichi Manyama, Table of n, a(n) for n = 0..1668
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Ângela Mestre and José Agapito, A Family of Riordan Group Automorphisms, J. Int. Seq., Vol. 22 (2019), Article 19.8.5.

See A034807 and A115149 for comments.

For convolutions of this sequence see A115141-A115149.

O.g.f.: 1/c(x) = 1-x*c(x) with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers).
a(0) = 1, a(n) = -C(n-1), n>=1, with C(n):=A000108(n) (Catalan).
G.f.: (1 + sqrt(1-4*x))/2=U(0) where U(k)=1 - x/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 29 2012
G.f.: 1/G(0) where G(k) = 1 - x/(x - 1/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 12 2012
G.f.: G(0), where G(k)= 2*x*(2*k+1) + k + 1 - 2*x*(k+1)*(2*k+3)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Jul 14 2013
D-finite with recurrence n*a(n) +2*(-2*n+3)*a(n-1)=0. a(n) = A002420(n)/2, n>0. - R. J. Mathar, Aug 09 2015
a(n) ~ -2^(2*n-2) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, May 06 2021

A116382 Riordan array (1/sqrt(1-4x^2), (1-2x^2c(x^2))(x^2c(x^2))/(x(1-x-x^2*c(x^2)))) where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 0, 3, 2, 1, 6, 4, 5, 3, 1, 0, 10, 10, 8, 4, 1, 20, 15, 21, 19, 12, 5, 1, 0, 35, 42, 42, 32, 17, 6, 1, 70, 56, 84, 92, 77, 50, 23, 7, 1, 0, 126, 168, 192, 180, 131, 74, 30, 8, 1, 252, 210, 330, 405, 400, 326, 210, 105, 38, 9, 1

Offset: 0

Paul Barry, Feb 12 2006

Row sums are A116383. Diagonal sums are A116384.
First column has e.g.f. Bessel_I(0,2*x) (A000984 with interpolated zeros).
Second column has e.g.f. Bessel_I(1,2*x) + Bessel_I(2,2*x) (A037952).
Third column has e.g.f. Bessel_I(2,2*x) + 2*Bessel_I(3,2*x) + Bessel_I(4,2*x) (A116385).
A binomial-Bessel triangle: column k has e.g.f. Sum_{j=0..k} C(k,j) * Bessel_I(k+j,2*x).

			Triangle begins
    1;
    0,   1;
    2,   1,   1;
    0,   3,   2,   1;
    6,   4,   5,   3,   1;
    0,  10,  10,   8,   4,   1;
   20,  15,  21,  19,  12,   5,   1;
    0,  35,  42,  42,  32,  17,   6,   1;
   70,  56,  84,  92,  77,  50,  23,   7,  1;
    0, 126, 168, 192, 180, 131,  74,  30,  8, 1;
  252, 210, 330, 405, 400, 326, 210, 105, 38, 9, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Flat(List([0..10], n-> List([0..n], k-> Sum([0..n], j-> (-1)^(n-j)*Binomial(n,j)*Sum([0..j], m-> Binomial(j,m-k)*Binomial(m,j-m)  ))))); # G. C. Greubel, May 22 2019

T:= func< n,k | (&+[(-1)^(n-j)*Binomial(n,j)*(&+[Binomial(j,m-k)* Binomial(m,j-m): m in [0..j]]): j in [0..n]]) >;
[[T(n,k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 22 2019

T[n_, k_] := Sum[(-1)^(n-j)*Binomial[n, j]*Sum[Binomial[j, i-k]* Binomial[i, j-i], {i, 0, j}], {j, 0, n}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 24 2018 *)

{T(n,k) = sum(j=0,n, (-1)^(n-j)*binomial(n,j)*sum(m=0,j, binomial(j,m-k)*binomial(m,j-m) ))}; \\ G. C. Greubel, May 22 2019

def T(n, k): return sum((-1)^(n-j)*binomial(n,j)*sum(binomial(j,m-k)*binomial(m,j-m) for m in (0..j)) for j in (0..n))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 22 2019

Riordan array (1/sqrt(1-4*x^2), sqrt(1-4*x^2)*(1-sqrt(1-4*x^2))/(x-2*x^2 + x*sqrt(1-4*x^2))).

Number triangle T(n,k) = Sum{j=0..n} (-1)^(n-j)* C(n,j)*Sum_{i=0..j} C(j,i-k)*C(i,j-i).

A121839 Decimal expansion of Sum_{k>=1} 1/C(k), where C(k) is a Catalan Number (A000108).

Original entry on oeis.org

1, 8, 0, 6, 1, 3, 3, 0, 5, 0, 7, 7, 0, 7, 6, 3, 4, 8, 9, 1, 5, 2, 9, 2, 3, 6, 7, 0, 0, 6, 3, 1, 8, 0, 3, 2, 5, 4, 5, 9, 5, 8, 4, 9, 9, 9, 1, 5, 2, 3, 2, 9, 1, 4, 4, 6, 9, 7, 7, 2, 6, 6, 3, 7, 9, 5, 0, 2, 7, 6, 9, 6, 9, 3, 8, 9, 4, 9, 0, 6, 1, 4, 9, 7, 0, 7, 2, 2, 2, 1, 6, 9, 8, 3, 1, 3, 7, 8, 5, 2, 8, 2

Offset: 1

Alexander Adamchuk, Aug 28 2006

			1.806133050770763489152923670063180325459584999152...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Alexander Adamchuk's post, Mathematics in Russian, August 29 2006.
Eric Weisstein's World of Mathematics, Catalan Number.

Cf. A000108, A002390, A268813 (essentially the same).

SetDefaultRealField(RealField(100)); R:=RealField(); 1 + 4*Sqrt(3)*Pi(R)/27; // G. C. Greubel, Nov 04 2018

evalf(1 + Sum((-1)^n*(2*n+1)/(9*n*(n+1)/2+1), n=0..infinity), 120); # Vaclav Kotesovec, May 31 2015

RealDigits[N[Sum[n!(n + 1)!/(2n)!, {n, 1, Infinity}], 150]]
RealDigits[N[1+4*Sqrt[3]*Pi/27,100]][[1]]

default(realprecision,100); 1 + 4*sqrt(3)*Pi/27

Reciprocal Catalan Constant C = 1 + 4*sqrt(3)*Pi/27.
This number is f(1) where f(x) = -1 + 2*(sqrt(4-x)*(8+x) + 12 * sqrt(x) * arctan(sqrt(x)/sqrt(4-x))) / sqrt((4-x)^5). This form corresponds to a generating function of the reciprocal Catalan numbers in the sense of Sprugnoli. - Juan M. Marquez, Mar 05 2009
Equals -1 + hypergeom([1,2],[1/2],1/4); note hypergeom([1,2],[1/2],x/4) = 1/1 + 1/1*x + 1/2*x^2 + 1/5*x^3 + 1/14*x^4 + 1/42*x^5 + ... is the g.f. for the inverse Catalan numbers (including C(0)). - Joerg Arndt, Apr 06 2013
From Vaclav Kotesovec, May 31 2015: (Start)
Equals 1 + Integral_{x=0..1} Product_{k>=1} (1-x^(9*k))^3 dx.
Equals 1 + Sum_{n>=0} (-1)^n * (2*n+1) / (9*n*(n+1)/2 + 1).
(End)
Equals 1 + Integral_{0..inf} x^3 BesselI_0(x) BesselK_0(x)^2 dx. - Jean-François Alcover, Jun 06 2016
From Amiram Eldar, Jul 05 2020: (Start)
Equals 1 + gamma(4/3)*gamma(5/3).
Equals 1 + Integral_{x=0..oo} dx/(1 + x^3)^2. (End)

A201204 Half-convolution of Catalan sequence A000108 with itself.

Original entry on oeis.org

1, 1, 3, 7, 23, 66, 227, 715, 2529, 8398, 30275, 104006, 380162, 1337220, 4939443, 17678835, 65844845, 238819350, 895451117, 3282060210, 12374186318, 45741281820, 173257703723, 644952073662, 2452607696798, 9183676536076, 35042725663002, 131873975875180, 504697422982484, 1907493251046152

Offset: 0

Wolfdieter Lang, Jan 02 2012

In general the half-convolution of a sequence {b(n)}_0^infty with itself is defined by chat(n):=sum(b(k)*b(n-k), k=0..floor(n/2)), n>=0. The o.g.f. of the sequence {chat(n)} is obtained from the bisection 2*chat(2*k) - b(k)^2 = c(2*k), k>=0, with the ordinary convolution c(n):=sum(b(k)*b(n-k),k=0..n), n>=0, and 2*chat(2*k+1) = c(2*k+1), k>=0. This leads to the o.g.f.s  for the corresponding even (e) and odd (o) parts:
  2*Chate(x) - B2(x) = Ce(x) and 2*Chato(x) = Co(x), where Chate(x):= sum(chat(2*k)*x^k,k=0..infty), Chato(x):= sum(chat(2*k+1)*x^k,k=0..infty), B2(x) := sum(b(k)^2*x^k, k=0..infty), Ce(x) := sum(c(2*k)*x^k, k=0..infty) and Co(x) := sum(c(2*k+1)*x^k, k=0..infty). Thus Chate(x)=(Ce(x) + B2(x))/2 and Chato(x)=Co(x)/2. Expressing this in terms of C(x), the o.g.f. of {c(n)}, and B2(x) leads to the result: Chat(x)= (C(x) + B2(x^2))/2.
In the Catalan case b(n)=A000108(n), c(n)=b(n+1), C(x)= (cata(x)+1)/x, with the o.g.f. of A000108 cata(x)=(1-sqrt(1-4*x))/(2*x), and B2(x) is found under A001246 to be (-1 + hypergeom([-1/2,-1/2],[1],16*x))/(4*x). This produces the o.g.f. given in the formula section.
This computation was motivated by a question about the o.g.f. of A000992 ("half-Catalan numbers"). Note, however, that this sequence is not the half-convolution of the Catalan numbers presented here.
Apparently the number of hills to the left of or at the midpoint in all Dyck paths of semilength n+1. [David Scambler, Apr 30 2013]

Robert Israel, Table of n, a(n) for n = 0..170

A000108, bisection: A201205 and A065097.

C:= n -> binomial(2*n,n)/(n+1):
A:= n -> add(C(k)*C(n-k),k=0..floor(n/2));
seq(A(i),i=1..100); # Robert Israel, Jun 06 2014

Table[Sum[CatalanNumber[k]CatalanNumber[n-k],{k,0,Floor[n/2]}],{n,0,30}] (* Harvey P. Dale, Jun 12 2012 *)
Table[CatalanNumber[n + 1]/2 + 2^(2 n + 1) Binomial[1/2, n/2 + 1]^2, {n, 0, 30}] (* Vladimir Reshetnikov, Oct 03 2016 *)

a(n) = sum(Catalan(k)*Catalan(n-k),k=0..floor(n/2)), n>=0, with Catalan(n)=A000108(n).
O.g.f.: G(x)=(catalan(x)-1)/(2*x)+(-1+hypergeom([-1/2,-1/2],[1],16*x^2))/(8*x^2), with the o.g.f. catalan(x) of the Catalan numbers (see also the comment section).
a(n) ~ 2^(2*n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 15 2014
a(n) = A000108(n+1)/2 + 2^(2*n+1) * binomial(1/2, n/2+1)^2. - Vladimir Reshetnikov, Oct 03 2016
D-finite with recurrence: (n+1)*(n+2)^2*a(n) +6*(n-2)*(n+1)^2*a(n-1) +4*(-16*n^3+25*n^2+4*n-4)*a(n-2) +16*(-4*n^3+25*n^2-56*n+41)*a(n-3) +192*(4*n-7)*(n-3)^2*a(n-4) -256*(2*n-7)*(n-4)^2*a(n-5)=0. - R. J. Mathar, Feb 21 2020

cp's OEIS Frontend

A131428 a(n) = 2*C(n) - 1, where C(n) = A000108(n) are the Catalan numbers.

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A076025 Expansion of g.f.: (1-3*x*C)/(1-4*x*C) where C = (1 - sqrt(1-4*x))/(2*x) = g.f. for Catalan numbers A000108.

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A076035 G.f.: 1/(1-4*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.

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A129442 Expansion of c(x)*c(x*c(x)) where c(x) is the g.f. of A000108.

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A187357 Catalan trisection: A000108(3*n), n >= 0.

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A025225 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 2. Also a(n) = (2^n)*C(n-1), where C = A000108 (Catalan numbers).

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A076025 Expansion of g.f.: (1-3xC)/(1-4xC) where C = (1 - sqrt(1-4x))/(2x) = g.f. for Catalan numbers A000108.

A076035 G.f.: 1/(1-4xC) where C = (1/2-1/2(1-4x)^(1/2))/x = g.f. for Catalan numbers A000108.

A129442 Expansion of c(x)c(xc(x)) where c(x) is the g.f. of A000108.

A025225 a(n) = a(1)a(n-1) + a(2)a(n-2) + ...+ a(n-1)a(1) for n >= 2. Also a(n) = (2^n)C(n-1), where C = A000108 (Catalan numbers).

A116382 Riordan array (1/sqrt(1-4x^2), (1-2x^2c(x^2))(x^2c(x^2))/(x(1-x-x^2*c(x^2)))) where c(x) is the g.f. of A000108.