A003168 -id:A003168 - cp's OEIS frontend

A361242 Number of nonequivalent noncrossing cacti with n nodes up to rotation.

1, 1, 1, 2, 7, 26, 144, 800, 4995, 32176, 215914, 1486270, 10471534, 75137664, 547756650, 4047212142, 30255934851, 228513227318, 1741572167716, 13380306774014, 103542814440878, 806476983310180, 6318519422577854, 49769050291536486, 393933908000862866

Offset: 0

Andrew Howroyd, Mar 07 2023

nonn

A noncrossing cactus is a connected noncrossing graph (A007297) that is a cactus graph (a tree of edges and polygons).

Since every cactus is an outerplanar graph, every cactus has at least one drawing as a noncrossing graph.

			The a(3) = 2 nonequivalent cacti have the following blocks:
   {{1,2}, {1,3}},
   {{1,2,3}}.
Graphically these can be represented:
        1           1
      /  \        /  \
     2    3      2----3
.
The a(4) = 7 nonequivalent cacti have the following blocks:
  {{1,2}, {1,3}, {1,4}},
  {{1,2}, {1,3}, {3,4}},
  {{1,2}, {1,4}, {2,3}},
  {{1,2}, {2,4}, {3,4}},
  {{1,2}, {1,3,4}},
  {{1,2}, {2,3,4}},
  {{1,2,3,4}}.
Graphically these can be represented:
   1---4    1   4    1---4    1   4
   | \      | \ |    |        | / |
   2   3    2   3    2---3    2   3
.
   1---4    1   4    1---4
   | \ |    | / |    |   |
   2   3    2---3    2---3

Andrew Howroyd, Table of n, a(n) for n = 0..500
Wikipedia, Cactus graph.
Index entries for sequences related to cacti.

Cf. A003168, A007297, A361236, A361243.

\\ Here F(n) is the g.f. of A003168.
F(n) = {1 + serreverse(x/((1+2*x)*(1+x)^2) + O(x*x^n))}
seq(n) = {my(f=F(n-1)); Vec(1 + intformal(f) - sum(d=2, n, eulerphi(d) * log(1-subst(x*f^2 + O(x^(n\d+1)),x,x^d)) / d), -n-1)}

A219536 G.f. satisfies A(x) = 1 + x(A(x)^2 + 2A(x)^3).

Original entry on oeis.org

1, 3, 24, 255, 3102, 40854, 566934, 8164263, 120864390, 1827982362, 28122626760, 438720097638, 6923868098820, 110346550539780, 1773394661610258, 28707809007278775, 467677404522668742, 7661583171651546786, 126137791939032756960, 2085923447593966281378

Offset: 0

Paul D. Hanna, Nov 21 2012

			G.f.: A(x) = 1 + 3*x + 24*x^2 + 255*x^3 + 3102*x^4 + 40854*x^5 +...
Related expansions:
A(x)^2 = 1 + 6*x + 57*x^2 + 654*x^3 + 8310*x^4 + 112560*x^5 +...
A(x)^3 = 1 + 9*x + 99*x^2 + 1224*x^3 + 16272*x^4 + 227187*x^5 +...
The g.f. satisfies A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where
G(x) = 1 + 3*x + 15*x^2 + 93*x^3 + 645*x^4 + 4791*x^5 +...+ A103210(n)*x^n +...

G. C. Greubel, Table of n, a(n) for n = 0..790
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Column k=2 of A336574.

Cf. A003168, A103210, A219534, A219535.

CoefficientList[1/x*InverseSeries[Series[4*x^2/(1-x-Sqrt[1-10*x+x^2]), {x, 0, 20}], x],x] (* Vaclav Kotesovec, Dec 28 2013 *)

/* Formula A(x) = 1 + x*(A(x)^2 + 2*A(x)^3): */
{a(n)=my(A=1);for(i=1,n,A=1+x*(A^2+2*A^3) +x*O(x^n));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

/* Formula using Series Reversion: */
{a(n)=my(A=1,G=(1-x-sqrt(1-10*x+x^2+x^3*O(x^n)))/(4*x));A=(1/x)*serreverse(x/G);polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

a(n) = sum(k=0, n, 2^k*binomial(n, k)*binomial(2*n+k+1, n)/(2*n+k+1)); \\ Seiichi Manyama, Jul 26 2020

a(n) = sum(k=0, n, 2^(n-k)*binomial(2*n+1, k)*binomial(3*n-k, n-k))/(2*n+1); \\ Seiichi Manyama, Jul 26 2020

Let G(x) = (1-x - sqrt(1 - 10*x + x^2)) / (4*x), then g.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion(x/G(x)),
(2) A(x) = G(x*A(x)) and G(x) = A(x/G(x)),
where G(x) is the g.f. of A103210.
Recurrence: 4*n*(2*n+1)*(19*n-26)*a(n) = (2717*n^3 - 6435*n^2 + 4342*n - 840)*a(n-1) + 2*(n-2)*(2*n-3)*(19*n-7)*a(n-2). - Vaclav Kotesovec, Dec 28 2013
a(n) ~ (3/19)^(1/4) * (5+sqrt(57)) * ((143 + 19*sqrt(57))/16)^n / (16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Dec 28 2013
From Seiichi Manyama, Jul 26 2020: (Start)
a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(2*n+k+1,n)/(2*n+k+1).
a(n) = (1/(2*n+1)) * Sum_{k=0..n} 2^(n-k) * binomial(2*n+1,k) * binomial(3*n-k,n-k). (End)
From Seiichi Manyama, Aug 10 2023: (Start)
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * 3^(n-k) * binomial(n,k) * binomial(3*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 3^k * 2^(n-k) * binomial(n,k) * binomial(2*n,k-1) for n > 0. (End)
a(n) = (-1)^(n+1) * (3/n) * Jacobi_P(n-1, 1, n+1, -5) for n >= 1. - Peter Bala, Sep 08 2024

A086456 Expansion of (1 + x + sqrt(1 - 6*x + x^2))/2 in powers of x.

Original entry on oeis.org

1, -1, -2, -6, -22, -90, -394, -1806, -8558, -41586, -206098, -1037718, -5293446, -27297738, -142078746, -745387038, -3937603038, -20927156706, -111818026018, -600318853926, -3236724317174, -17518619320890, -95149655201962

Offset: 0

Michael Somos, Jul 20 2003

sign

Series reversion of x(Sum_{k>=0} a(k)x^k) is x(Sum_{k>=0} A003168(k)x^k).

G.f. A(x) = Sum_{k>=0} a(k)x^k satisfies 0 = 2*x - (x + 1)*A(x) + A(x)^2.

Foissy, Loic, Algebraic structures on double and plane posets, J. Algebr. Comb. 37, No. 1, 39-66 (2013).

A minor variation of A006318. a(n)=-A006318(n-1), n>0. a(n)=A085403(n), n>1.

Cf. A001003.

ReciprocalSeries[ser_, n_] := CoefficientList[ Series[1/ser, {x, 0, n}], x];
LittleSchroeder := (1 + x - Sqrt[1 - 6 x + x^2])/(4 x); (* A001003 *)
ReciprocalSeries[LittleSchroeder, 22] (* Peter Luschny, Jan 10 2019 *)

a(n)=polcoeff((1+x+sqrt(1-6*x+x^2+x*O(x^n)))/2,n)

G.f.: (1 + x + sqrt(1 - 6*x + x^2))/2. (= 1/g.f. A001003)

D-finite with recurrence: n*a(n) + 3*(-2*n + 3)*a(n-1) + (n-3)*a(n-2) = 0. - R. J. Mathar, Jul 23 2017

A185047 Expansion of 2F1( [1, 4/3]; [3]; 9*x).

Original entry on oeis.org

1, 4, 21, 126, 819, 5616, 40014, 293436, 2200770, 16805880, 130245570, 1021926780, 8102419470, 64819355760, 522606055815, 4242331511910, 34645707347265, 284459491903860, 2346790808206845, 19444838125142430, 161745698950048395, 1350224965148230080

Offset: 0

Olivier Gérard, Feb 15 2011

Close to A003168.

Can be seen as a degree 3 analog of the Catalan numbers A000108 (which would be degree 2).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000108, A003168, A034164.

A185047:=n->-9^(n+1)*binomial(n+1/3, n+2): seq(A185047(n), n=0..30); # Wesley Ivan Hurt, Feb 16 2017

CoefficientList[Series[ HypergeometricPFQ[{1, 4/3}, {3}, 9 x], {x, 0, 20}], x]

a(n) = -9^(n+1)*binomial(n+1/3, n+2). - Karol A. Penson, Nov 06 2015
a(n) = (1/(6*sqrt(3)*Pi))*Integral_{x = 0..9} x^n*x^(1/3)*(9 - x)^(2/3). Cf. A034164. - Peter Bala, Nov 17 2015
O.g.f.: (1 - (1-9*x)^(2/3) - 6*x)/(9*x^2).
D-finite with recurrence (n+2)*a(n) +3*(-3*n-1)*a(n-1)=0. - R. J. Mathar, Jul 27 2022
Sum_{n>=0} 1/a(n) = (3/512)*(92 + 15*Pi*sqrt(3) + 45*log(3)). - Amiram Eldar, Dec 18 2022
a(n) = ((n + 3)/3) * Product_{1 <= i <= j <= n} (2*i + j + 3)/(2*i + j - 1). - Peter Bala, Feb 22 2023

A243662 Triangle read by rows: the reversed x = 1+q Narayana triangle at m=2.

Original entry on oeis.org

1, 3, 1, 12, 8, 1, 55, 55, 15, 1, 273, 364, 156, 24, 1, 1428, 2380, 1400, 350, 35, 1, 7752, 15504, 11628, 4080, 680, 48, 1, 43263, 100947, 92169, 41895, 9975, 1197, 63, 1, 246675, 657800, 708400, 396704, 123970, 21560, 1960, 80, 1, 1430715, 4292145, 5328180, 3552120, 1381380, 318780, 42504, 3036, 99, 1

Offset: 1

N. J. A. Sloane, Jun 13 2014

See Novelli-Thibon (2014) for precise definition.
From Tom Copeland, Dec 13 2022: (Start)
The row polynomials are the nonvanishing numerator polynomials generated in the compositional, or Lagrange, inversion in x about the origin of the odd o.g.f. Od1(x,t) = x*(t*(1-x^2)-x^2) / (1-x^2) = t*x - x^3 - x^5 - x^7 - x^9 - ... .
For example, from the Lagrange inversion formula (LIF), the tenth derivative in x of (x/Od1(x,t))^11 / 11! = (1/((t*(1-x^2)-x^2) / (1-x^2)))^11 / 11! at x = 0 is (t^4 + 24*t^3 + 156*t^2 + 364*t + 273) / t^16. These polynomials are also generated by the iterated derivatives ((1/(D Od1(x,t)) D)^n g(x) evaluated at x = 0 where D = d/dx.
An explicit generating function for the polynomials can be obtained by finding the solution of the cubic equation y - t*x - y*x^2 + (1+t)*x^3 = 0 for x in terms of y and t that satisfies y(x=0;t) = 0 = x(y=0;t).
The row polynomials are also the polynomials generated in the compositional inverse of O(x,t) = x / (1+(1+t)x)*(1+x)^2) = x + (-t - 3)*x^2 + (t^2 + 4 t + 6)*x^3 + (-t^3 - 5*t^2 - 10*t - 10)*x^4 + ..., containing the truncated Pascal polynomials of A104712 / A325000.
For example, from the LIF, the third derivative of ((1 + (1+t)*x)*(1+x)^2)^4 / 4! at x = 0 is 55 + 55*t + 15*t^2 + t^3.
A natural refinement of this array was provided in a letter by Isaac Newton in 1676--a set of partition polynomials for generating the o.g.f. of the compositional inverse of the generic odd o.g.f. x + u_1 x^3 + u_2 x^5 + ... in the infinite set of indeterminates u_n. (End)
T(n,k) is the number of noncrossing cacti with n+1 nodes and n+1-k blocks. See A361242. - Andrew Howroyd, Apr 13 2023

			Triangle begins:
     1;
     3,    1;
    12,    8,    1;
    55,   55,   15,   1;
   273,  364,  156,  24,  1;
  1428, 2380, 1400, 350, 35, 1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened)
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Fig. 10.

Cf. A001764, A001263, A243663 (m=3).
Row sums give A003168.
Row reversed triangle is A102537.

T[m_][n_, k_] := Binomial[(m + 1) n + 1 - k, n - k] Binomial[n, k - 1]/n;
Table[T[2][n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 12 2019 *)

T(n)=[Vecrev(p) | p<-Vec(serreverse(x/((1+x+x*y)*(1+x)^2) + O(x*x^n)))]
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Apr 13 2023

T(n,k) = (binomial(3*n+1,n) * binomial(n,k-1) * binomial(n-1,k-1)) / (binomial(3*n,k-1) * (3*n+1)) = (A001764(n) * A001263(n,k) * k) / binomial(3*n,k-1) for 1 <= k <= n (conjectured). - Werner Schulte, Nov 22 2018
T(n,k) = binomial(3*n+1-k,n-k) * binomial(n,k-1) / n for 1 <= k <= n, more generally: T_m(n,k) = binomial((m+1)*n+1-k,n-k) * binomial(n,k-1) / n for 1 <= k <= n and some fixed integer m > 1. - Werner Schulte, Nov 22 2018
G.f.: A(x,y) is the series reversion of x/((1 + x + x*y)*(1 + x)^2). - Andrew Howroyd, Apr 13 2023

Data and Example (T(2,2) and T(5,3)) corrected and more terms added by Werner Schulte, Nov 22 2018

A290722 Number of dissections of a 2n-gon into polygons with even number of sides counted up to rotations and reflections.

Original entry on oeis.org

1, 2, 4, 13, 48, 238, 1325, 8297, 54519, 373363, 2621872, 18797682, 136969519, 1011903735, 7564219361, 57129086391, 435394899361, 3345082819597, 25885718422329, 201619294539406, 1579629974876090, 12442262963919863, 98483477967355109, 783017782731507416

Offset: 2

Evgeniy Krasko, Sep 03 2017

nonn

			For a(4) = 4 the dissections of an octagon are: two dissections into 3 quadrangles; a dissection into one hexagon and one quadrangle; a dissection into one octagon.

Andrew Howroyd, Table of n, a(n) for n = 2..200
Evgeniy Krasko, Alexander Omelchenko, Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees, The Electronic Journal of Combinatorics, 22 (2015), #P1.17.

Cf. A003168 (counted distinctly).

Cf. A001004, A290816, A295419.

\\ See A295419 for DissectionsModDihedral().
select(v->v>0, DissectionsModDihedral(apply(v->v%2==0, [1..50]))) \\ Andrew Howroyd, Nov 22 2017

Terms a(8) and beyond from Andrew Howroyd, Nov 22 2017

A211788 Triangle enumerating certain two-line arrays of positive integers.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 7, 21, 21, 1, 10, 47, 126, 126, 1, 13, 82, 324, 818, 818, 1, 16, 126, 642, 2300, 5594, 5594, 1, 19, 179, 1107, 4977, 16741, 39693, 39693, 1, 22, 241, 1746, 9335, 38642, 124383, 289510, 289510, 1, 25, 312, 2586, 15941, 77273, 301630, 939880, 2157150, 2157150

Offset: 1

Peter Bala, Aug 02 2012

This is the table of f(n,k) in the notation of Carlitz (p.123). The triangle enumerates two-line arrays of positive integers
............a_1 a_2 ... a_n..........
............b_1 b_2 ... b_n..........
such that
1) max(a_i, b_i) <= min(a_(i+1), b_(i+1)) for 1 <= i <= n-1
2) max(a_i, b_i) <= i for 1 <= i <= n
3) a_n = b_n = k.
See A071948 and A193091 for other two-line array enumerations.
It appears that the row reverse array is the Riordan array (f(x), g(x)), where f(x) = 1 + x + 4*x^2 + 21*x^3 + 126*x^4 + 818*x^5 + ... is the g.f. of A003168 and g(x) = x + 3*x^2 + 14*x^3 + 79*x^4 + 494*x^5 + 3294*x^6 + ... is the g.f. of A003169. - Peter Bala, Nov 26 2024

			Triangle begins
.n\k.|..1....2....3....4....5....6
= = = = = = = = = = = = = = = = = =
..1..|..1
..2..|..1....1
..3..|..1....4....4
..4..|..1....7...21...21
..5..|..1...10...47..126..126
..6..|..1...13...82..324..818..818
...
T(4,2) = 7: The 7 two-line arrays are
...1 1 1 2....1 1 2 2....1 2 2 2....1 1 1 2
...1 1 1 2....1 1 2 2....1 2 2 2....1 1 2 2
...........................................
...1 1 2 2....1 1 2 2....1 2 2 2...........
...1 1 1 2....1 2 2 2....1 1 2 2...........

L. Carlitz, Enumeration of two-line arrays, Fib. Quart., Vol. 11 Number 2 (1973), 113-130.

Cf. A003168 (main diagonal), A211789 (row sums).

Cf. A003169, A071948, A193091.

Recurrence equation:
T(1,1) = 1; T(n,n) = T(n,n-1); T(n+1,k) = Sum_{j = 1..k} (2*k-2*j+1)*T(n,j) for 1 <= k <= n.
T(n+1,k+1) = (1/n) * ((n - k)*Sum_{i = 0..k} C(n, k-i)*C(2*n+i, i) + Sum_{i = 1..k} C(n, k-i)*C(2*n+i, i-1)).
Row reverse has production matrix
  1 1
  3 3 1
  5 5 3 1
  7 7 5 3 1
  ...
Main diagonal T(n,n) = A003168(n). Row sums A211789.

A259554 a(n) = Sum_{i=0..n} (2^(i)(-1)^(i+n)C(n,i)C(2n+i-1,n-1)).

Original entry on oeis.org

1, 7, 52, 403, 3206, 25954, 212738, 1760035, 14666470, 122920642, 1035046816, 8749594462, 74207078908, 631140253072, 5381022869822, 45975731083555, 393556869530630, 3374504760608026, 28977403637496104, 249167023897718138

Offset: 1

Vladimir Kruchinin, Jun 30 2015

G. C. Greubel, Table of n, a(n) for n = 1..1000
V. V. Kruchinin and D. V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.

Cf. A002002, A003168.

a := n -> n*hypergeom([2*n+1, -n+1], [2], -1):
seq(simplify(a(n)), n=1..9); # Peter Luschny, Oct 07 2016

Table[Sum[2^i * (-1)^(i+n) * Binomial[n, i] * Binomial[2*n+i-1, n-1], {i, 0, n}], {n,1,20}] (* Vaclav Kotesovec, Jul 01 2015 *)

a(n):=sum(2^(i)*(-1)^(i+n)*binomial(n,i)*binomial(2*n+i-1,n-1),i,0,n);

a(n) = sum(i=0, n, 2^i*(-1)^(i+n)*binomial(n, i)*binomial(2*n+i-1, n-1)); \\ Michel Marcus, Jul 02 2015

G.f.: A(x) = x*B(x)'/B(x), where B(x) is g.f. of A003168.
Recurrence: 4*n*(2*n-1)*(17*n^2 - 51*n + 38)*a(n) = (1207*n^4 - 4828*n^3 + 6659*n^2 - 3662*n + 672)*a(n-1) - 2*(n-2)*(2*n-3)*(17*n^2 - 17*n + 4)*a(n-2). - Vaclav Kotesovec, Jul 01 2015
a(n) ~ (71+17*sqrt(17))^n / (17^(1/4) * sqrt(Pi*n) * 2^(4*n+1)). - Vaclav Kotesovec, Jul 01 2015
a(n) = (1/2)*Sum_{k = 0..n} binomial(n-1,n-k)*binomial(2*n+k-1,k). - Vladimir Kruchinin, Oct 07 2016
a(n) = n*hypergeom([2*n+1, -n+1], [2], -1). - Peter Luschny, Oct 07 2016
From Peter Bala, Nov 08 2022: (Start)
a(n) = (1/2)*[x^n] ( (1 - x)/(1 - 2*x) )^(2*n). Cf. A002002(n) = [x^n] ( (1 - x)/(1 - 2*x) )^n.
a(n) = (1/2)*Sum_{k = 0..n} (-1)^(n-k)*2^k*binomial(2*n,n-k)*binomial(2*n+k-1,k).
a(n) = (1/2)*(-1)^n*binomial(2*n,n)*hypergeom( [-n, 2*n], [n+1], 2).
The Gauss congruences hold: a(n*p^r) == a(n^p^(r-1)) (mod p^r) for all primes p >= 3 and all positive integers n and r. (End)

A361243 Number of nonequivalent noncrossing cacti with n nodes up to rotation and reflection.

Original entry on oeis.org

1, 1, 1, 2, 5, 17, 79, 421, 2537, 16214, 108204, 743953, 5237414, 37574426, 273889801, 2023645764, 15128049989, 114256903169, 870786692493, 6690155544157, 51771411793812, 403238508004050, 3159259746188665, 24884525271410389, 196966954270163612

Offset: 0

Andrew Howroyd, Mar 07 2023

nonn

A noncrossing cactus is a connected noncrossing graph (A007297) that is a cactus graph (a tree of edges and polygons).

			The a(4) = 5 nonequivalent cacti have the following blocks:
  {{1,2}, {1,3}, {1,4}},
  {{1,2}, {1,3}, {3,4}},
  {{1,2}, {1,4}, {2,3}},
  {{1,2}, {1,3,4}},
  {{1,2,3,4}}.
Graphically these can be represented:
   1---4    1   4    1---4    1---4    1---4
   | \      | \ |    |        | \ |    |   |
   2   3    2   3    2---3    2   3    2---3

Andrew Howroyd, Table of n, a(n) for n = 0..500
Wikipedia, Cactus graph.
Index entries for sequences related to cacti.

Cf. A003168, A007297, A361239, A361242.

\\ Here F(n) is the g.f. of A003168.
F(n) = {1 + serreverse(x/((1+2*x)*(1+x)^2) + O(x*x^n))}
seq(n) = {my(f=F(n-1)); Vec(1/(1 - x*subst(f + O(x^(n\2+1)), x, x^2)) + 1 + intformal(f) - sum(d=2, n, eulerphi(d) * log(1-subst(x*f^2 + O(x^(n\d+1)),x,x^d)) / d), -n-1)/2}

A100328 Column 1 of triangle A100326, in which row n equals the inverse binomial of column n of square array A100324, with leading zero omitted.

Original entry on oeis.org

1, 4, 20, 116, 736, 4952, 34716, 250868, 1855520, 13979192, 106901032, 827644424, 6474611984, 51100656544, 406400018092, 3253636464756, 26201323746880, 212093247874904, 1724793778005528, 14084738953391768, 115447965121881856

Offset: 0

Paul D. Hanna, Nov 17 2004

Self-convolution of A100327, which equals the row sums of triangle A100326.

Vaclav Kotesovec, Table of n, a(n) for n = 0..650

Cf. A003168, A003169, A100324, A100326, A100327.

{a(n)=if(n==0,1,sum(j=0,n, if(j==0,1,sum(k=0,j,2*binomial(j,k)*binomial(2*j+k,k-1)/j))* if(n-j==0,1,sum(k=0,n-j,2*binomial(n-j,k)*binomial(2*n-2*j+k,k-1)/(n-j)))))}

G.f.: (1+G003169(x))*G003169(x)/x, where G003169(x) is the g.f. of A003169.
Recurrence: 4*(n+1)*(2*n+1)*(17*n^2 - 28*n + 12)*a(n) = (1207*n^4 - 1988*n^3 + 1013*n^2 - 124*n - 12)*a(n-1) - 2*(n-2)*(2*n-3)*(17*n^2 + 6*n + 1)*a(n-2). - Vaclav Kotesovec, Jul 05 2014
a(n) ~ sqrt(95+393/sqrt(17)) * ((71+17*sqrt(17))/16)^n / (4*sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 05 2014
From Peter Bala, Sep 08 2024: (Start)
a(n) = (2/n) * Sum_{k = 0..n} binomial(n+1, n-k-1)*binomial(2*n, k)*2^(n-k) for n >= 1.
a(n) = 4*Jacobi_P(n-1, 2, n+1, 3)/n for n >= 1. Cf. A003168. (End)

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