A006013 -id:A006013 - cp's OEIS frontend

A024492 Catalan numbers with odd index: a(n) = binomial(4n+2, 2n+1)/(2*n+2).

1, 5, 42, 429, 4862, 58786, 742900, 9694845, 129644790, 1767263190, 24466267020, 343059613650, 4861946401452, 69533550916004, 1002242216651368, 14544636039226909, 212336130412243110, 3116285494907301262, 45950804324621742364, 680425371729975800390

Offset: 0

Clark Kimberling

a(n) and Catalan(n) have the same 2-adic valuation (equal to 1 less than the sum of the digits in the binary representation of (n + 1)). In particular, a(n) is odd iff n is of the form 2^m - 1. - Peter Bala, Aug 02 2016

			sqrt((1/2)*(1+sqrt(1-x))) = 1 - (1/8)*x - (5/128)*x^2 - (42/2048)*x^3 - ...

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, Advances in Applied Mathematics, Vol. 121 (2020), 102105; arXiv preprint, arXiv:1907.10725 [math.CO], 2019.
Pedro J. Miana and Natalia Romero, Moments of combinatorial and Catalan numbers, Journal of Number Theory, Vol. 130, No. 8 (August 2010), pp. 1876-1887. See Remark 3, p. 1882.

Cf. A048990 (Catalan numbers with even index), A024491, A000108, A000894.

Cf. A000260, A006013, A187357, A187358, A187359, A228329, A276484.

[Factorial(4*n+2)/(Factorial(2*n+1)*Factorial(2*n+2)): n in [0..20]]; // Vincenzo Librandi, Sep 13 2011

with(combstruct):bin := {B=Union(Z,Prod(B,B))}: seq (count([B,bin,unlabeled],size=2*n), n=1..18); # Zerinvary Lajos, Dec 05 2007
a := n -> binomial(4*n+1, 2*n+1)/(n+1):
seq(a(n), n=0..17); # Peter Luschny, May 30 2021

CoefficientList[ Series[1 + (HypergeometricPFQ[{3/4, 1, 5/4}, {3/2, 2}, 16 x] - 1), {x, 0, 17}], x]
CatalanNumber[Range[1,41,2]] (* Harvey P. Dale, Jul 25 2011 *)

a(n):=sum((k+1)^2*binomial(2*(n+1),n-k)^2,k,0,n)/(n+1)^2; /* Vladimir Kruchinin, Oct 14 2014 */

combinat::catalan(2*n+1)$ n = 0..24 // Zerinvary Lajos, Jul 02 2008

combinat::dyckWords::count(2*n+1)$ n = 0..24 // Zerinvary Lajos, Jul 02 2008

a(n)=binomial(4*n+2,2*n+1)/(2*n+2) \\ Charles R Greathouse IV, Sep 13 2011

G.f.: (1/2)*x^(-1)*(1-sqrt((1/2)*(1+sqrt(1-16*x)))).
G.f.: 3F2([3/4, 1, 5/4], [3/2, 2], 16*x). - Olivier Gérard, Feb 16 2011
a(n) = 4^n*binomial(2n+1/2, n)/(n+1). - Paul Barry, May 10 2005
a(n) = binomial(4n+1,2n+1)/(n+1). - Paul Barry, Nov 09 2006
a(n) = (1/(2*Pi))*Integral_{x=-2..2} (2+x)^(2*n)*sqrt((2-x)*(2+x)). - Peter Luschny, Sep 12 2011
D-finite with recurrence (n+1)*(2*n+1)*a(n) -2*(4*n-1)*(4*n+1)*a(n-1)=0. - R. J. Mathar, Nov 26 2012
G.f.: (c(sqrt(x)) - c(-sqrt(x)))/(2*sqrt(x)) = (2-(sqrt(1-4*sqrt(x)) + sqrt(1+4*sqrt(x))))/(4*x), with the g.f. c(x) of the Catalan numbers A000108. - Wolfdieter Lang, Feb 23 2014
a(n) = Sum_{k=0..n} (k+1)^2*binomial(2*(n+1),n-k)^2 /(n+1)^2. - Vladimir Kruchinin, Oct 14 2014
G.f.: A(x) = (1/x)*(inverse series of x - 5*x^2 + 8*x^3 - 4*x^4). - Vladimir Kruchinin, Oct 31 2014
a(n) ~ sqrt(2)*16^n/(sqrt(Pi)*n^(3/2)). - Ilya Gutkovskiy, Aug 02 2016
Sum_{n>=0} 1/a(n) = A276484. - Amiram Eldar, Nov 18 2020
G.f.: A(x) = C(4*x)*C(x*C(4*x)), where C(x) is the g.f. of A000108. - Alexander Burstein, May 01 2021
a(n) = (1/Pi)*16^(n+1)*Integral_{x=0..Pi/2} (cos x)^(4n+2)*(sin x)^2. - Greg Dresden, May 30 2021
Sum_{n>=0} a(n)/4^n = 2 - sqrt(2). - Amiram Eldar, Mar 16 2022
From Peter Bala, Feb 22 2023: (Start)
a(n) = (1/4^n) * Product_{1 <= i <= j <= 2*n} (i + j + 2)/(i + j - 1).
a(n) = Product_{1 <= i <= j <= 2*n} (3*i + j + 2)/(3*i + j - 1). Cf. A000260. (End)

More terms from Wolfdieter Lang

A022340 Even Fibbinary numbers (A003714); also 2*Fibbinary(n).

Original entry on oeis.org

0, 2, 4, 8, 10, 16, 18, 20, 32, 34, 36, 40, 42, 64, 66, 68, 72, 74, 80, 82, 84, 128, 130, 132, 136, 138, 144, 146, 148, 160, 162, 164, 168, 170, 256, 258, 260, 264, 266, 272, 274, 276, 288, 290, 292, 296, 298, 320, 322, 324, 328, 330, 336, 338, 340, 512

Offset: 0

Marc LeBrun

nonn

Positions of ones in binomial(3k+2,k+1)/(3k+2) modulo 2 (A085405). - Paul D. Hanna, Jun 29 2003
Construction: start with strings S(0)={0}, S(1)={2}; for k>=2, concatenate all prior strings excluding S(k-1) and add 2^k to each element in the resulting string to obtain S(k); this sequence is the concatenation of all such generated strings: {S(0),S(1),S(2),...}. Example: for k=5, concatenate {S(0),S(1),S(2),S(3)} = {0, 2, 4, 8,10}; add 2^5 to each element to obtain S(5)={32,34,38,40,42}. - Paul D. Hanna, Jun 29 2003
From Gus Wiseman, Apr 08 2020: (Start)
The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. This sequence lists all numbers k such that the k-th composition in standard order has no ones. For example, the sequence together with the corresponding compositions begins:
()          80: (2,5)        260: (6,3)
(2)         82: (2,3,2)      264: (5,4)
(3)         84: (2,2,3)      266: (5,2,2)
(4)        128: (8)          272: (4,5)
(2,2)      130: (6,2)        274: (4,3,2)
(5)        132: (5,3)        276: (4,2,3)
(3,2)      136: (4,4)        288: (3,6)
(2,3)      138: (4,2,2)      290: (3,4,2)
(6)        144: (3,5)        292: (3,3,3)
(4,2)      146: (3,3,2)      296: (3,2,4)
(3,3)      148: (3,2,3)      298: (3,2,2,2)
(2,4)      160: (2,6)        320: (2,7)
(2,2,2)    162: (2,4,2)      322: (2,5,2)
(7)        164: (2,3,3)      324: (2,4,3)
(5,2)      168: (2,2,4)      328: (2,3,4)
(4,3)      170: (2,2,2,2)    330: (2,3,2,2)
(3,4)      256: (9)          336: (2,2,5)
(3,2,2)    258: (7,2)        338: (2,2,3,2)
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Equals 2 * A003714.
Cf. A006013, A001045, A085405, A085407.
Compositions with no ones are counted by A212804.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Compositions without terms > 2 are A003754.
- Compositions without ones are A022340 (this sequence).
- Sum is A070939.
- Compositions with no twos are A175054.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Runs-resistance is A333628.
Cf. A066099, A124767, A228351, A318928, A333218.

a022340 = (* 2) . a003714 -- Reinhard Zumkeller, Feb 03 2015

f[n_Integer] := Block[{k = Ceiling[ Log[ GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k-- ]; FromDigits[fr, 2]]; Select[f /@ Range[0, 95], EvenQ[ # ] &] (* Robert G. Wilson v, Sep 18 2004 *)
Select[Range[2, 512, 2], BitAnd[#, 2#] == 0 &] (* Alonso del Arte, Jun 18 2012 *)

from itertools import count, islice
def A022340_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:not n&(n>>1),count(max(0,startvalue+(startvalue&1)),2))
A022340_list = list(islice(A022340_gen(),30)) # Chai Wah Wu, Sep 07 2022

def A022340(n):
    tlist, s = [1,2], 0
    while tlist[-1]+tlist[-2] <= n: tlist.append(tlist[-1]+tlist[-2])
    for d in tlist[::-1]:
        if d <= n:
            s += 1
            n -= d
        s <<= 1
    return s # Chai Wah Wu, Apr 24 2025

For n>0, a(F(n))=2^n, a(F(n)-1)=A001045(n+2)-1, where F(n) is the n-th Fibonacci number with F(0)=F(1)=1.

a(n) + a(n)/2 = a(n) XOR a(n)/2, see A106409. - Reinhard Zumkeller, May 02 2005

Edited by Ralf Stephan, Sep 01 2004

A055113 Number of bracketings of 0^0^0^...^0, with n 0's, giving the result 0, with conventions that 0^0 = 1^0 = 1^1 = 1, 0^1 = 0.

Original entry on oeis.org

0, 1, 0, 1, 1, 5, 11, 41, 120, 421, 1381, 4840, 16721, 59357, 210861, 759071, 2744393, 10000437, 36609977, 134750450, 498016753, 1848174708, 6882643032, 25715836734, 96365606679, 362102430069, 1364028272451, 5150156201026, 19486989838057, 73880877535315

Offset: 0

Jeppe Stig Nielsen, Jun 15 2000

Total number of bracketings of 0^0^...^0 is A000108(n-1) (this is Catalan's problem). So the number of bracketings giving 1 is A000108(n-1) - a(n).
Also bracketings of f => f => ... => f where f is "false" and "=>" is implication.
Self-convolution yields A187430. - Paul D. Hanna, May 31 2015
Also, number of nonnegative walks of n steps with step sizes 1 and 2, starting at 0 and ending at 1. - Andrew Howroyd, Dec 23 2017
Series reversion is related to A001006. - F. Chapoton, Jul 14 2021

			Number of bracketings of 0^0^0^0^0^0 giving 0 is 11, so a(6) = 11.
From _Jon E. Schoenfield_, Dec 24 2017: (Start)
The 11 ways of parenthesizing 0^0^0^0^0^0 to obtain 0 are
0^(0^(0^((0^0)^0))) = 0^(0^(0^(1^0))) = 0^(0^(0^1)) = 0^(0^0) = 0^1 = 0;
0^((0^0)^(0^(0^0))) = 0^(1^(0^1)) = 0^(1^0) = 0^1 = 0;
0^((0^0)^((0^0)^0)) = 0^(1^(1^0)) = 0^(1^1) = 0^1 = 0;
0^(((0^0)^0)^(0^0)) = 0^((1^0)^1) = 0^(1^1) = 0^1 = 0;
0^((0^(0^(0^0)))^0) = 0^((0^(0^1))^0) = 0^((0^0)^0) = 0^(1^0) = 0^1 = 0;
0^((0^((0^0)^0))^0) = 0^((0^(1^0))^0) = 0^((0^1)^0) = 0^(0^0) = 0^1 = 0;
0^(((0^0)^(0^0))^0) = 0^((1^1)^0) = 0^(1^0) = 0^1 = 0;
0^(((0^(0^0))^0)^0) = 0^(((0^1)^0)^0) = 0^((0^0)^0) = 0^(1^0) = 0^1 = 0;
0^((((0^0)^0)^0)^0) = 0^(((1^0)^0)^0) = 0^((1^0)^0) = 0^(1^0) = 0^1 = 0;
(0^(0^0))^((0^0)^0) = (0^1)^(1^0) = 0^1 = 0;
(0^((0^0)^0))^(0^0) = (0^(1^0))^1. (End)

Thanks to Soren Galatius Smith, Jesper Torp Kristensen et al.

Alois P. Heinz, Table of n, a(n) for n = 0..1671 (terms n = 1..200 from T. D. Noe)
E. A. Bender and S. G. Williamson, Foundations of Combinatorics with Applications (see Chap. 11, Example 11.3, pp. 312-313 and Example 11.31, pp. 351-352).
V. Čačić and V. Kovač, On the fraction of IL formulas that have normal forms, arXiv preprint arXiv:1309.3408 [math.LO], 2013-2015.
D. G. Rogers, Comments on A111160, A055113 and A006013
Wikipedia, Zero to the power of zero

Column k=2 of A185286.

Cf. A000108, A001006, A055392, A055395, A006632, A111160, A187430, A296619, A306668.

a:= proc(n) option remember; `if`(n<3, n*(2-n),
      ((n-1)*(115*n^3-689*n^2+1332*n-840) *a(n-1)
       +(8*n-20)*(5*n^3+12*n^2-113*n+126) *a(n-2)
       -36*(n-3)*(5*n-8)*(2*n-5)*(2*n-7)  *a(n-3))
      /((2*(2*n-1))*(5*n-13)*n*(n-1)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 04 2019

Rest[ CoefficientList[ Series[(-1 - Sqrt[1 - 4x] + Sqrt[2]Sqrt[1 + Sqrt[1 - 4x] + 6x])/4, {x, 0, 28}], x]] (* Robert G. Wilson v, Oct 28 2005 *)
a[n_] := (-1)^(n+1)*Binomial[2n-1, n]*HypergeometricPFQ[{1-n, (n+1)/2, n/2}, {n, n+1}, 4]/(2n-1);
Array[a, 27] (* Jean-François Alcover, Dec 26 2017, after Vladimir Kruchinin *)

a(n):= sum(binomial(2*j+n-1,j+n-1)*(-1)^(n-j-1)*binomial(2*n-1,j+n), j,0,n-1)/(2*n-1); /* Vladimir Kruchinin, May 10 2011 */

a(n)={sum(j=0, n-1, binomial(2*j+n-1, j+n-1)*(-1)^(n-j-1)*binomial(2*n-1, j+n))/(2*n-1)} \\ Andrew Howroyd, Dec 23 2017

first(n) = x='x+O('x^(n+1)); Vec(-((1 - 4*x)^(1/2) + 1)/4 + (2 + 2*(1 - 4*x)^(1/2) + 12*x)^(1/2)/4) \\ Iain Fox, Dec 23 2017

G.f.: - 1/4 - (1/4)*(1 - 4*x)^(1/2) + (1/4)*(2 + 2*(1 - 4*x)^(1/2) + 12*x)^(1/2).
The ratio a(n)/A000108(n-1) converges to (5-sqrt(5))/10 as n->oo.
a(n) = (Sum_{j=0..n-1} binomial(2*j+n-1, j+n-1)*(-1)^(n-j-1)*binomial(2*n-1, j+n))/(2*n-1). - Vladimir Kruchinin, May 10 2011
a(n) ~ (1-1/sqrt(5))*2^(2*n-3)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 09 2013
D-finite with recurrence: 2*n*(2*n-1)*(n-1)*a(n) -(n-1)*(19*n^2-60*n+48)*a(n-1) +(-31*n^3+173*n^2-346*n+264)*a(n-2) +4*(2*n-7)*(17*n^2-83*n+102)*a(n-3) +36*(2*n-7)*(2*n-9)*(n-4)*a(n-4)=0. - R. J. Mathar, Feb 20 2020
a(n) + A111160(n) = A000108(n). - F. Chapoton, Jul 14 2021

a(0)=0 prepended by Alois P. Heinz, Mar 04 2019

A004319 a(n) = binomial(3*n, n - 1).

Original entry on oeis.org

1, 6, 36, 220, 1365, 8568, 54264, 346104, 2220075, 14307150, 92561040, 600805296, 3910797436, 25518731280, 166871334960, 1093260079344, 7174519270695, 47153358767970, 310325523515700, 2044802197953900, 13488561475572645, 89067326568860640, 588671286046028640

Offset: 1

N. J. A. Sloane

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Milan Janjic, Two Enumerative Functions.
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.

Cf. A045721 (k=1), A025174 (k=2), A236194 (k=4), A013698 (k=5), A165817 (k=-1), A117671 (k=-2).

Cf. A000139, A001764, A005809, A006013, A236194, A001791, A004331, A004343, A004356, A004369, A004382.

A004319 := proc(n)
binomial(3*n,n-1);
end proc: # R. J. Mathar, Aug 10 2015

Table[Binomial[3n, n - 1], {n, 20}] (* Harvey P. Dale, Sep 21 2011 *)

a(n):=sum((binomial(3*i-1,2*i-1)*binomial(3*n-3*i-3,2*n-2*i-2))/(2*n-2*i-1),i,1,n-1)/2; /* Vladimir Kruchinin, May 15 2013 */

vector(30, n, binomial(3*n, n-1)) \\ Altug Alkan, Nov 04 2015

G.f.: (g-1)/(1-3*z*g^2), where g = g(z) is given by g = 1 + z*g^3, g(0) = 1, i.e. (in Maple notation), g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z). - Emeric Deutsch, May 22 2003
a(n) = Sum_{i=0..n-1} binomial(i+2*n, i). - Ralf Stephan, Jun 03 2005
D-finite with recurrence -2*(2*n+1)*(n-1)*a(n) + 3*(3*n-1)*(3*n-2)*a(n-1) = 0. - R. J. Mathar, Feb 05 2013
a(n) = (1/2) * Sum_{i=1..n-1} binomial(3*i - 1, 2*i - 1)*binomial(3*n - 3*i - 3, 2*n - 2*i - 2)/(2*n - 2*i - 1). - Vladimir Kruchinin, May 15 2013
G.f.: x*hypergeom2F1(5/3, 4/3; 5/2; 27x/4). - R. J. Mathar, Aug 10 2015
a(n) = n*A001764(n). - R. J. Mathar, Aug 10 2015
From Peter Bala, Nov 04 2015: (Start)
With offset 0, the o.g.f. equals f(x)*g(x)^3, where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k, n). See the cross-references. (End)
G.f.: cos(t)/(2*sqrt(1 - (27*x)/4)) - sin(t)/(sqrt(3)*sqrt(x)), where t = arcsin((sqrt(27*x))/2)/3. - Vladimir Kruchinin, May 13 2016
a(n) = [x^(2*n+1)] 1/(1 - x)^n. - Ilya Gutkovskiy, Oct 10 2017
a(n) = binomial(n+1, 2) * A000139(n). - F. Chapoton, Feb 23 2024

A307678 G.f. A(x) satisfies: A(x) = 1 + x*A(x)^3/(1 - x).

Original entry on oeis.org

1, 1, 4, 19, 101, 578, 3479, 21714, 139269, 912354, 6078832, 41066002, 280636657, 1936569717, 13475408847, 94446518559, 666149216744, 4724705621702, 33676421377532, 241100485812034, 1732999323835918, 12501487280292424, 90478497094713958, 656788523782034248, 4780725762185300389

Offset: 0

Ilya Gutkovskiy, Apr 21 2019

nonn

Convolution square root of A270386.

			G.f.: A(x) = 1 + x + 4*x^2 + 19*x^3 + 101*x^4 + 578*x^5 + 3479*x^6 + 21714*x^7 + 139269*x^8 + 912354*x^9 + 6078832*x^10 + ...

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Guillermo Esteban, Clemens Huemer, and Rodrigo I. Silveira, New production matrices for geometric graphs, arXiv:2003.00524 [math.CO], 2020.

Cf. A001764, A002212, A006013, A127897, A188687 (partial sums), A270386.

terms = 24; A[] = 1; Do[A[x] = 1 + x A[x]^3/(1 - x) + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[Sum[a[k] a[i - k] a[j - i], {k, 0, i}], {i, 0, j}], {j, 0, n - 1}]; Table[a[n], {n, 0, 24}]
terms = 24; CoefficientList[Series[2 Sqrt[(1 - x) Sin[1/3 ArcSin[3/2 Sqrt[3] Sqrt[x/(1 - x)]]]^2/x]/Sqrt[3], {x, 0, terms}], x]

a(n):=sum(binomial(n-1,n-k)*(binomial(3*k,k))/(2*k+1),k,0,n); /* Vladimir Kruchinin, Feb 05 2022*/

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^3, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

a(0) = 1; a(n) = Sum_{j=0..n-1} Sum_{i=0..j} Sum_{k=0..i} a(k)*a(i-k)*a(j-i).
a(n) ~ 31^(n + 1/2) / (3*sqrt(Pi) * n^(3/2) * 2^(2*n+2)). - Vaclav Kotesovec, May 06 2019
G.f.: (2/sqrt(3*x/(1-x)))*sin((1/3)*asin(sqrt((27*x/(1-x))/4))). - Vladimir Kruchinin, Feb 05 2022
a(n) = Sum_{k=0..n} C(n-1,n-k)*C(3*k,k)/(2*k+1). - Vladimir Kruchinin, Feb 05 2022

A258708 Triangle read by rows: T(i,j) = integer part of binomial(i+j, i-j)/(2*j+1) for i >= 1 and j = 0..i-1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 7, 4, 1, 1, 7, 14, 12, 5, 1, 1, 9, 25, 30, 18, 6, 1, 1, 12, 42, 66, 55, 26, 7, 1, 1, 15, 66, 132, 143, 91, 35, 8, 1, 1, 18, 99, 245, 334, 273, 140, 45, 9, 1, 1, 22, 143, 429, 715, 728, 476, 204, 57, 10, 1

Offset: 1

N. J. A. Sloane, Jun 12 2015

In the Loh-Shannon-Horadam paper, Table 3 contains a typo (see Extensions lines).
T(n,k) = round(A258993(n,k)/(2*k+1)). - Reinhard Zumkeller, Jun 22 2015
From Reinhard Zumkeller, Jun 23 2015: (Start)
(using tables 4 and 5 of the Loh-Shannon-Horadam paper, p. 8f).
T(n, n-1)  = 1;
T(n, n-2)  = n for n > 1;
T(n, n-3)  = A000969(n-3) for n > 2;
T(n, n-4)  = A000330(n-3) for n > 3;
T(n, n-5)  = T(2*n-7, 2) = A000970(n) for n > 4;
T(n, n-6)  = A000971(n) for n > 5;
T(n, n-7)  = A000972(n) for n > 6;
T(n, n-8)  = A000973(n) for n > 7;
T(n, 1)    = A001840(n-1) for n > 1;
T(2*n, n)  = A001764(n);
T(3*n-1, 1) = A000326(n);
T(3*n, 2*n) = A002294(n);
T(4*n, 3*n) = A002296(n). (End)

			Triangle T(i, j) (with rows i >= 1 and columns j >= 0) begins as follows:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,   1;
  1,  5,  7,   4,   1;
  1,  7, 14,  12,   5,   1;
  1,  9, 25,  30,  18,   6,   1;
  1, 12, 42,  66,  55,  26,   7,  1;
  1, 15, 66, 132, 143,  91,  35,  8, 1;
  1, 18, 99, 245, 334, 273, 140, 45, 9, 1;
  ...

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.

Cf. A011793.
Cf. A007318, A258993, A158405, A005408, A006013 (central terms).
Cf. A000326, A000330, A000969, A000970, A000971, A000972, A000973, A000976, A001764, A001840, A002294, A002296.

a258708 n k = a258708_tabl !! (n-1) !! k
a258708_row n = a258708_tabl !! (n-1)
a258708_tabl = zipWith (zipWith ((round .) . ((/) `on` fromIntegral)))
                       a258993_tabl a158405_tabl
-- Reinhard Zumkeller, Jun 22 2015, Jun 16 2015

Corrected T(8,5) = 26 from Reinhard Zumkeller, Jun 13 2015

A129872 Sequence M_n arising in enumeration of arrays of directed blocks (see 2007 Quaintance reference for precise definition). [The next term is not an integer.].

Original entry on oeis.org

1, 4, 16, 72, 364, 1916, 10581, 59681, 343903, 2010089

Offset: 1

N. J. A. Sloane, May 26 2007

nonn

Warning: as defined the terms are not integral in general: 1, 4, 16, 72, 364, 1916, 10581, 59681, 343903, 2010089, 23798969/2, ... - Jocelyn Quaintance, Mar 31 2013.

Jocelyn Quaintance, Letter Representations of m x n x p Proper Arrays (2004), arXiv:math/0412244. See Table 1.
Jocelyn Quaintance, Combinatoric Enumeration of Two-Dimensional Proper Arrays, Discrete Math., 307 (2007), 1844-1864. See Table 1.

Cf. A129873-A129886.

listrn(m) = {R = t*O(t); for (n= 1, m, R = (2*t^11 + t^10 + 3*t^9 + 8*t^8 + 12*t^7 + 16*t^6 + 26*t^5 + 20*t^4 + 16*t^3 + 7*t^2 + t - (t^14 - t^12 - 8*t^10 - 11*t^8 - 5*t^6 + t^4)*R^3 - (3*t^13 + t^12 - 5*t^11 - 2*t^10 - 34*t^9 - 14*t^8 - 39*t^7 - 16*t^6 - 11*t^5 - 2*t^4 + 6*t^3 + 2*t^2)*R^2)/(t^12 + t^11 - 7*t^10 - 7*t^9 - 42*t^8 - 35*t^7 - 51*t^6 - 41*t^5 - 14*t^4 - 4*t^3 + 8*t^2 + 6*t + 1);); return(vector(m, i , polcoeff(R, i, t)));}
listbn(m) = {B = t*O(t); for (n= 1, m, B = (1 + 2*t*B^2 - t^2*B^3 );); return(vector(m, i , polcoeff(B, i, t)));}
listMn(m) = {b = listbn(m); /* see also A006013 */ s = listsn(m); /* see A129873 */ d = listdn(m); /* see A129880 */ r = listrn(m); for (i=1, m, v = (b[i] + s[i] - d[i] - r[i])/2; print1(v, ", "););}
\\ Michel Marcus, Mar 30 2013

See Quaintance reference for generating functions that produce A129872-A129886.

Edited by N. J. A. Sloane, Nov 29 2016

A013698 a(n) = binomial(3*n+2, n-1).

Original entry on oeis.org

1, 8, 55, 364, 2380, 15504, 100947, 657800, 4292145, 28048800, 183579396, 1203322288, 7898654920, 51915526432, 341643774795, 2250829575120, 14844575908435, 97997533741800, 647520696018735, 4282083008118300, 28339603908273840, 187692294101632320, 1243918491347262900

Offset: 1

Joachim.Rosenthal(AT)nd.edu (Joachim Rosenthal) and Emeric Deutsch

Degree of variety K_{2,n}^1. Also number of double-rises (or odd-level peaks) in all generalized {(1,2),(1,-1)}-Dyck paths of length 3(n+1).
Number of dissections of a convex (2n+2)-gon by n-2 noncrossing diagonals into (2j+2)-gons, 1<=j<=n-1.
a(n) is the number of lattice paths from (0,0) to (3n+1,n-1) avoiding two consecutive up-steps. - Shanzhen Gao, Apr 20 2010

Robert Israel, Table of n, a(n) for n = 1..1000
M. S. Ravi, Joachim Rosenthal, and Xiaochang Wang, Dynamic Pole Assignment and Schubert Calculus, SIAM J. Control Optimization, 34 (1996), 813-832, esp. p. 825.
Daniel W. Stasiuk, An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads, Master's Thesis, University of Saskatchewan-Saskatoon (2018).

Cf. A013699 (q=2), A013700 (q=3), A013701 (q=4), A013702 (q=5).

A column of triangle A102537. Cf. A001764, A004319, A005809, A006013, A025174, A045721, A117671, A165817, A236194.

List([1..25], n-> Binomial(3*n+2, n-1)) # G. C. Greubel, Mar 21 2019

[Binomial(3*n+2, n-1): n in [1..25]]; // Vincenzo Librandi, Aug 10 2015

seq(binomial(3*n+2,n-1), n=0..30); # Robert Israel, Aug 09 2015

Table[Binomial[3*n+2, n-1], {n, 25}] (* Arkadiusz Wesolowski, Apr 02 2012 *)

first(m)=vector(m,n,binomial(3*n+2, n-1)); /* Anders Hellström, Aug 09 2015 */

[binomial(3*n+2, n-1) for n in (1..25)] # G. C. Greubel, Mar 21 2019

G.f.: g/((g-1)^3*(3*g-1)) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 09 2011
a(n) = Sum_{k=0..n-1} binomial(2*n+k+2,k). - Arkadiusz Wesolowski, Apr 02 2012
D-finite with recurrence 2*(2*n+3)*(n+1)*a(n) - n*(67*n+34)*a(n-1) + 30*(3*n-1)*(3*n-2)*a(n-2) = 0. - R. J. Mathar, Feb 05 2013
a(n+1) = (3*n+5)*(3*n+4)*(3*n+3)*a(n)/((2*n+5)*(2*n+4)*n). - Robert Israel, Aug 09 2015
With offset 0, the o.g.f. equals f(x)*g(x)^5, where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k,n). Cf. A045721 (k = 1), A025174 (k = 2), A004319 (k = 3), A236194 (k = 4), A165817 (k = -1), A117671 (k = -2). - Peter Bala, Nov 04 2015
a(n) ~ 3^(3*n+5/2) / (4^(n+2) * sqrt(Pi*n)). - Amiram Eldar, Sep 05 2025

A054727 Number of forests of rooted trees with n nodes on a circle without crossing edges.

Original entry on oeis.org

1, 2, 7, 33, 181, 1083, 6854, 45111, 305629, 2117283, 14929212, 106790500, 773035602, 5652275723, 41683912721, 309691336359, 2315772552485, 17415395593371, 131632335068744, 999423449413828, 7618960581522348, 58295017292748756, 447517868947619432, 3445923223190363608

Offset: 1

Philippe Flajolet, Apr 20 2000

Andrew Howroyd, Table of n, a(n) for n = 1..200
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).
Guillermo Esteban, Clemens Huemer, Rodrigo I. Silveira, New production matrices for geometric graphs, arXiv:2003.00524 [math.CO], 2020.
Philippe Flajolet, Source
Philippe Flajolet, Enumeration of planar configurations in computational geometry
P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 486, 502
Samuele Giraudo, Combalgebraic structures on decorated cliques, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 8, arXiv:1709.08416 [math.CO], 2017.
Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.

Row sums of A094021.

Cf. A006013, A000108.

ZZ:=[F,{F=Union(Epsilon,ZB),ZB=Prod(Z1,P),P=Sequence(B),B=Prod(P,Z1,P),Z1=Prod(Z,F)}, unlabeled]: seq(count(ZZ,size=n),n=1..20); # Zerinvary Lajos, Apr 22 2007

a[n_] := (3*n-3)!/((n-1)!*(2*n-1)!)*HypergeometricPFQ[{1-2*n, 1-n, -n}, {3/2 - 3*n/2, 2 - 3*n/2}, -1/4]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Sep 05 2012, after formula *)

N=33; x='x+O('x^N); Vec(serreverse(x/((1+x)*(1-sqrt(1-4*x))/(2*x)))) \\ Joerg Arndt, May 25 2016

a(n) = Sum_{j=1..n} binomial(n, j-1) * binomial(3*n-2*j-1, n-j) / (2*n - j).
G.f. A(x) satisfies 2*A(x)^2=x*(1-sqrt(1-4*A(x)))*(1-A(x)). - Vladimir Kruchinin, Nov 25 2011
From Peter Bala, Nov 07 2015: (Start)
O.g.f. A(x) = revert(x/((1 + x)*C(x))), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f for the Catalan numbers A000108.
Row sums of A094021. (End)
Conjecture: -2(37*n-80) *(n-1) *(2*n-1) *a(n) +2*(592*n^3-3056*n^2+5045*n-2665) *a(n-1) +2*(148*n^3-986*n^2+2021*n-1255) *a(n-2) -5*(n-5) *(n-2) *(37*n-43) *a(n-3)=0. - R. J. Mathar, Apr 30 2018
a(n) ~ c * d^n / (sqrt(Pi) * n^(3/2)), where d = 8.2246915409778560686084627753... is the real root of the equation 5 - 8*d - 32*d^2 + 4*d^3 = 0 and c = 0.07465927842190452347018812862935237... is the positive real root of the equation -125 + 22376*c^2 + 8880*c^4 + 592*c^6 = 0. - Vaclav Kotesovec, Apr 30 2018

A005156 Number of alternating sign 2n+1 X 2n+1 matrices symmetric about the vertical axis (VSASM's); also 2n X 2n off-diagonally symmetric alternating sign matrices (OSASM's).

Original entry on oeis.org

1, 1, 3, 26, 646, 45885, 9304650, 5382618660, 8878734657276, 41748486581283118, 559463042542694360707, 21363742267675013243931852, 2324392978926652820310084179576, 720494439459132215692530771292602232, 636225819409712640497085074811372777428304

Offset: 0

N. J. A. Sloane

a(n+1) is the Hankel transform of A006013. - Paul Barry, Jan 20 2007

a(n+1) is the Hankel transform of A025174(n+1). - Paul Barry, Apr 14 2008

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 201, VS(2n+1).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Gheorghe Coserea, Table of n, a(n) for n = 0..66
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 23.
M. T. Batchelor, J. de Gier and B. Nienhuis, The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions, arXiv:cond-mat/0101385 [cond-mat.stat-mech], 2001, (see A_V(2n+1)).
N. T. Cameron, Random walks, trees and extensions of Riordan group techniques, Dissertation, Howard University, 2002.
J. de Gier, Loops, matchings and alternating-sign matrices, arXiv:math/0211285 [math.CO], 2002-2003.
I. Fischer, The number of monotone triangles with prescribed bottom row, arXiv:math/0501102 [math.CO], 2005.
I. Gessel and G. Xin, The generating function of ternary trees and continued fractions, arXiv:math/0505217 [math.CO], 2005.
W. Hebsich and M. Rubey, Extended Rate, More Gfun, arXiv:math/0702086 [math.CO], 2007. [See p. 23.]
G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001, (see A_V(2n+1)).
A. V. Razumov and Yu. G. Stroganov, On refined enumerations of some symmetry classes of alternating sign matrices, arXiv:math-ph/0312071, 2003.
D. P. Robbins, Symmetry classes of alternating sign matrices, arXiv:math/0008045 [math.CO], 2000.
R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.
R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]

Cf. A109074, A134357.

A005156 := proc(n) local i,j,t1; (-3)^(n^2)*mul( mul( (6*j-3*i+1)/(2*j-i+2*n+1), j=1..n ),i=1..2*n+1); end;

Table[1/2^n Product[((6k-2)!(2k-1)!)/((4k-1)!(4k-2)!),{k,n}],{n,0,20}] (* Harvey P. Dale, Jul 07 2011 *)

a(n) = prod(k = 0, n-1, (3*k+2)*(6*k+3)!*(2*k+1)!/((4*k+2)!*(4*k+3)!));
vector(15, n, a(n-1))  \\ Gheorghe Coserea, May 30 2016

The formula for a(n) (see the Maple code) was conjectured by Robbins and proved by Kuperberg.
a(n) = (1/2^n) * Product_{k=1..n} ((6k-2)!(2k-1)!)/((4k-1)!(4k-2)!) (Razumov/Stroganov).
a(n) ~ exp(1/72) * Pi^(1/6) * 3^(3*n^2 + 3*n/2 + 11/72) / (A^(1/6) * GAMMA(1/3)^(1/3) * n^(5/72) * 2^(4*n^2 + 3*n + 1/9)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 01 2015

cp's OEIS Frontend

A024492 Catalan numbers with odd index: a(n) = binomial(4*n+2, 2*n+1)/(2*n+2).

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A022340 Even Fibbinary numbers (A003714); also 2*Fibbinary(n).

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A055113 Number of bracketings of 0^0^0^...^0, with n 0's, giving the result 0, with conventions that 0^0 = 1^0 = 1^1 = 1, 0^1 = 0.

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A004319 a(n) = binomial(3*n, n - 1).

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A307678 G.f. A(x) satisfies: A(x) = 1 + x*A(x)^3/(1 - x).

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A258708 Triangle read by rows: T(i,j) = integer part of binomial(i+j, i-j)/(2*j+1) for i >= 1 and j = 0..i-1.

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A024492 Catalan numbers with odd index: a(n) = binomial(4n+2, 2n+1)/(2*n+2).