A153231 -id:A153231 - cp's OEIS frontend

A367389 Expansion of g.f. A(x) = B(x^2)/(1 - 2xB(x^2)) where B(x) = 1 + 2xB(x)^3 is the g.f. of A153231.

1, 2, 6, 16, 52, 152, 512, 1568, 5392, 16992, 59232, 190336, 669952, 2183680, 7742464, 25512448, 90974464, 302368256, 1083175424, 3625435136, 13036688384, 43889186816, 158323564544, 535639556096, 1937483350016, 6582584115200, 23865932414976, 81381420826624, 295661476642816

Offset: 0

Paul D. Hanna, Dec 20 2023

nonn

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 16*x^3 + 52*x^4 + 152*x^5 + 512*x^6 + 1568*x^7 + 5392*x^8 + 16992*x^9 + 59232*x^10 + ...
where
1/A(x) = 1 - 2*x - 2*x^2 - 8*x^4 - 56*x^6 - 480*x^8 - 4576*x^10 - 46592*x^12 - ... - 2^n*binomial(3*n-1,n)/(3*n-1) * x^(2*n) - ...

Cf. A153231.

CoefficientList[1/(-2*x + x/InverseSeries[Series[x - 2*x^3, {x, 0, 30}], x]), x] (* Vaclav Kotesovec, Dec 24 2023 *)

{a(n) = my(A = 1/(-2*x + x/serreverse(x - 2*x^3 + O(x^(n+2))))); polcoeff(A,n)}
for(n=0, 30, print1(a(n), ", "))

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A;
A[#A] = polcoeff( (1 + (m+1)*x*Ser(A))^m / Ser(A)^m , m-1)/m - (m-1)^(m-2) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = B(x^2)/(1 - 2*x*B(x^2)) where B(x) = 1 + 2*x*B(x)^3 is the g.f. of A153231.
(2) A(x) = 1/(1-2*x - Sum_{n>=1} 2^n * binomial(3*n-1,n)/(3*n-1) * x^(2*n) ).
(3) A(x) = 1/(-2*x + x/Series_Reversion( x - 2*x^3 )).
(4) [x^(n-1)] (1 + (n+1)*x*A(x))^n / A(x)^n = n*(n-1)^(n-2) for n > 1.
(5) [x^(n-1)] (1 + n*x*A(x))^n / A(x)^n = -n*(n-2)^(n-2) for n > 1.
(6) [x^(n-1)] (1 + (n-2)*x*A(x))^n / A(x)^n = -n*(5*n-14)*(n-4)^(n-3) for n >= 1.
a(n) ~ (15*sqrt(3/2)/2 + 9 + (15*sqrt(3/2)/2 - 9)*(-1)^n) * 3^(3*n/2) / (sqrt(Pi) * n^(3/2) * 2^(n/2)). - Vaclav Kotesovec, Dec 24 2023

A349253 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - 2 * x * A(x)^2)).

Original entry on oeis.org

1, 3, 19, 169, 1753, 19795, 236035, 2923857, 37256881, 485202307, 6429346899, 86405569657, 1174917167881, 16134949855251, 223460304878467, 3117521211476641, 43771643214792033, 618045740600046211, 8770377489446217235, 125013010654218317385, 1789104455068153153849

Offset: 0

Ilya Gutkovskiy, Nov 12 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..845

Cf. A001764, A103210, A153231, A162326, A199475, A349254, A349255, A349256.

nmax = 20; A[] = 0; Do[A[x] = 1/((1 - x) (1 - 2 x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 1 + 2 Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 20}]
Table[Sum[Binomial[n + k, n - k] 2^k Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 20}]
a[n_] := HypergeometricPFQ[{1/3,2/3,-n,n + 1}, {1/2,1,3/2}, -(3/2)^3];
Table[a[n], {n, 0, 20}] (* Peter Luschny, Nov 12 2021 *)

a(n) = 1 + 2 * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) = Sum_{k=0..n} binomial(n+k,n-k) * 2^k * binomial(3*k,k) / (2*k+1).
a(n) = hypergeom([1/3, 2/3, -n, n + 1], [1/2, 1, 3/2], -(3/2)^3). - Peter Luschny, Nov 12 2021
a(n) ~ sqrt(315 + 31*sqrt(105)) * (31 + 3*sqrt(105))^n / (9 * sqrt(Pi) * 2^(2*n + 5/2) * n^(3/2)). - Vaclav Kotesovec, Nov 13 2021

A228484 a(n) = 2^n(3n)!/(n!(2n)!).

Original entry on oeis.org

1, 6, 60, 672, 7920, 96096, 1188096, 14883840, 188280576, 2399654400, 30766095360, 396363202560, 5126871859200, 66538909237248, 866061993246720, 11300615801536512, 147773778404769792, 1936073567335219200, 25408660721789829120, 333963051307735449600

Offset: 0

Johannes W. Meijer, Aug 22 2013

Oblique diagonal of the Pell-Jacobsthal triangle A013609. Its mirror diagonal is A006588.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A005809, A013609, A006588, A137209.

[2^n*Factorial(3*n)/(Factorial(n)*Factorial(2*n)): n in [0..20]]; // Vincenzo Librandi, Aug 24 2013

a := n -> 2^n*binomial(3*n, n): seq(a(n), n=0..16);

Table[2^n (3 n)!/(n! (2 n)!), {n, 0, 20}] (* Vincenzo Librandi, Aug 24 2013 *)

a(n) = 2^n*binomial(3*n, 2*n); \\ Michel Marcus, Mar 06 2022

a(n) = 2^n*A005809(n).
a(n) = A013609(3*n, n).
a(n) = A006588(n)/2^n.
a(n) = (2*n+1)*A153231(n).
Asymptotic approximation of a(n) ~ C*(13.5)^n/sqrt(n) with C = (1/2)*sqrt(3/Pi) = A137209.
Sum_{n>=0} 1/a(n) = (11*Pi - 12*log(2) + 270)/250. - Amiram Eldar, Mar 06 2022
From Karol A. Penson, Feb 26 2025: (Start)
G.f.:   hypergeom([1/3,2/3],[1/2],27*z/2).
E.g.f.: hypergeom([1/3,2/3],[1/2,1],27*z/2). (End)

More terms from Vincenzo Librandi, Aug 24 2013

A349255 G.f. A(x) satisfies A(x) = 1 / ((1 + x) * (1 - 2 * x * A(x)^2)).

Original entry on oeis.org

1, 1, 7, 47, 369, 3113, 27631, 254239, 2403361, 23201393, 227771831, 2266983119, 22822484497, 231994748633, 2377894546783, 24548520253247, 255026759000897, 2664111200687969, 27967731861910759, 294900120348032623, 3121862973452544433, 33167268461833410569

Offset: 0

Ilya Gutkovskiy, Nov 12 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..950

Cf. A001003, A001764, A153231, A337168, A346627, A349253, A349254, A349256.

nmax = 21; A[] = 0; Do[A[x] = 1/((1 + x) (1 - 2 x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = (-1)^n + 2 Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 21}]
Table[Sum[(-1)^(n - k) Binomial[n + k, n - k] 2^k Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 21}]
a[n_] := (-1)^n*HypergeometricPFQ[{1/3, 2/3, -n, n + 1}, {1/2, 1, 3/2}, (3/2)^3]; Table[a[n], {n, 0, 21}] (* Peter Luschny, Nov 12 2021 *)

a(n) = (-1)^n + 2 * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+k,n-k) * 2^k * binomial(3*k,k) / (2*k+1).
a(n) = (-1)^n*hypergeom([1/3, 2/3, -n, n + 1], [1/2, 1, 3/2], (3/2)^3). - Peter Luschny, Nov 12 2021
a(n) ~ sqrt(171 + 23*sqrt(57)) * (23 + 3*sqrt(57))^n / (9 * sqrt(Pi) * n^(3/2) * 2^(2*n + 5/2)). - Vaclav Kotesovec, Nov 13 2021

A364167 Expansion of g.f. A(x) satisfying A(x) = 1 + x * A(x)^3 * (1 + A(x)^3).

Original entry on oeis.org

1, 2, 18, 234, 3570, 59586, 1053570, 19392490, 367677090, 7131417282, 140834140722, 2822214963882, 57243994984722, 1172991472484610, 24245748916730658, 504935751379031082, 10584721220759172162, 223163804001804187266, 4729176407109705542994, 100676187744957784842090

Offset: 0

Seiichi Manyama, Jul 13 2023

Alois P. Heinz, Table of n, a(n) for n = 0..737 (first 50 terms from Christian N. Hofmann)

Cf. A144097, A153231, A363311.

a:= n-> sum(binomial(n, k)*binomial(3*n+3*k+1, n)/(3*n+3*k+1), k=0..n):
seq(a(n), n=0..49); # Christian N. Hofmann, Jul 14 2023

a(n) = sum(k=0, n, binomial(n, k)*binomial(3*n+3*k+1, n)/(3*n+3*k+1));

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(3*n+3*k+1,n)/(3*n+3*k+1).

A217364 a(n) = 2^nbinomial(5n, n)/(4*n+1).

Original entry on oeis.org

1, 2, 20, 280, 4560, 80960, 1520064, 29680640, 596593920, 12262581760, 256556410880, 5445566730240, 116974976102400, 2538140268625920, 55548588652625920, 1224777962595287040, 27180522769369333760, 606648758810237337600

Offset: 0

R. J. Mathar, Oct 01 2012

Old name was: Series reversion of x - 2*x^5.
Regular blocks of 3 intermediate zeros have been left out of the sequence: If y = x - 2x^5, then x = y + 2*y^5 + 20*y^9 + 280*y^13 + 4560*y^17 - ...
a(n) is the number of lattice paths (Schroeder paths) from (0,0) to (n,5n) with unit steps N=(0,1), E=(1,0) and D=(1,1) staying weakly above the line y = 5x with the total number of occurrences of NE and D equal to n. - Michael D. Weiner, Aug 21 2019

Alois P. Heinz, Table of n, a(n) for n = 0..724 (terms n=1..76 from Robert Israel)
D. Birmajer, J. B. Gil, J. D. Gil and M. D. Weiner, Schröder Coloring and Applications, arXiv:1908.08103 [math.CO], 2019.
R. J. Mathar, Series Expansion of Generalized Fresnel Integrals, arXiv:1211.3963 [math.CA], 2012.

Cf. A153231 (x-2*x^3), A217360 (x-2*x^4).

[2^n*Binomial(5*n, n)/(4*n+1): n in [0..25]]; // Michael D. Weiner, Aug 21 2019

A217364 := j -> pochhammer(1/5, j)*pochhammer(2/5, j)*pochhammer(3/5, j)*pochhammer(4/5, j)*6250^j/(4*j+1)!:
seq(A217364(n),n=0..100); # Robert Israel, May 19 2014

Table[CoefficientList[InverseSeries[Series[x-2*x^5,{x,0,100}],x],x][[4*n-2]],{n,1,20}] (* Vaclav Kotesovec,Aug 20 2013 *)
Table[2^n Binomial[5 n, n] / (4 n + 1), {n, 0, 20}] (* Michael D. Weiner, Aug 21 2019 *)

for(n=0,20, print1(round(if(n==0, 1, 5*2^n*gamma(5*n)/(gamma(n)*gamma(4*n+2)))), ", ")) \\ G. C. Greubel, Apr 01 2017

{my(x='x+O('x^133), v=Vec(serreverse(x - 2*x^5))); vector(#v\4, n, v[4*n-3] ) } \\ Joerg Arndt, Apr 02 2017

A217364 = lambda n: 5*2^n*gamma(5*n)/(gamma(n)*gamma(4*n+2)) if n > 0 else 1
[A217364(n) for n in (0..18)] # Peter Luschny, May 19 2014

D-finite with recurrence (4*n+1)*(4*n)*(4*n-1)*(4*n-2)*a(n) - 10*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-1) = 0.
a(n) ~ 5^(5*n)/(2^(7*n+4)*sqrt(Pi/10)*n^(3/2)). - Vaclav Kotesovec, Aug 20 2013
a(n) = Gamma(n+1/5)*Gamma(n+2/5)*Gamma(n+3/5)*Gamma(n+4/5)*6250^(n+1)*sqrt(5)/ (25000*Pi^2*Gamma(4*n+2)). - Robert Israel, May 19 2014
a(n) = 5*2^n*Gamma(5*n)/(Gamma(n)*Gamma(4*n+2)) for n > 0. - Peter Luschny, May 19 2014
G.f.: F([1/5, 2/5, 3/5, 4/5], [1/2, 3/4, 5/4], 3125*x/128), where F is the generalized hypergeometric function. - Stefano Spezia, Sep 03 2019
G.f. A(x) satisfies: A(x) = 1 / (1 - 2 * x * A(x)^4). - Ilya Gutkovskiy, Nov 12 2021

New definition and offset from Michael D. Weiner, Sep 03 2019

A348912 G.f. A(x) satisfies A(x) = (1 + 2 * x * A(x)^3) / (1 - x).

Original entry on oeis.org

1, 3, 21, 201, 2217, 26535, 335001, 4391553, 59203137, 815580507, 11430639165, 162470033625, 2336381642649, 33930648153615, 496935405133617, 7331179445170689, 108846406625097729, 1625145134034548019, 24385673680861258533, 367546405595389076649, 5561980053932228243529

Offset: 0

Ilya Gutkovskiy, Nov 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..828

Cf. A001764, A103210, A153231, A346626.

nmax = 20; A[] = 0; Do[A[x] = (1 + 2 x A[x]^3)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + 2 Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 20}]

a(n) = sum(k=0, n, 2^k*binomial(n, k)*binomial(n+2*k+1, n)/(n+2*k+1)); \\ Seiichi Manyama, Jul 24 2023

a(0) = 1; a(n) = a(n-1) + 2 * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) ~ sqrt(-50 + 30*sqrt(3) + (22 - 12*sqrt(3))*(2*(sqrt(3) - 1))^(1/3) + (2*(sqrt(3) - 1))^(2/3)*(-11 + 7*sqrt(3)))/(4*sqrt(3*Pi)*(-1 + sqrt(3))^(3/2) * n^(3/2) * (1 + (3*(-1 + sqrt(3))^(1/3))/2^(2/3) - 3/(2*(-1 + sqrt(3)))^(1/3))^n). - Vaclav Kotesovec, Nov 04 2021
a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(n+2*k+1,n) / (n+2*k+1). - Seiichi Manyama, Jul 24 2023

A217360 a(n) = 2^nbinomial(4n, n)/(3*n+1).

Original entry on oeis.org

1, 2, 16, 176, 2240, 31008, 453376, 6888960, 107707392, 1721477120, 28000141312, 461964898304, 7712495058944, 130050777006080, 2211737871974400, 37892693797109760, 653389823437701120, 11330548232319664128, 197475886172892823552

Offset: 0

R. J. Mathar, Oct 01 2012

Old name was: Series reversion of x - 2*x^4.
Regular blocks of 2 intermediate zeros have been removed from the sequence: If y = x - 2*x^4, then x = y + 2*y^4 + 16*y^7 + 176*y^10 + 2240*y^13 + 31008*y^16 + ...
a(n) is the number of lattice paths (Schroeder paths) from (0,0) to (n,4n) with unit steps N=(0,1), E=(1,0) and D=(1,1) staying weakly above the line y = 4x with the total number of occurrences of NE and D equal to n. - Michael D. Weiner, Jul 25 2019

Jinyuan Wang, Table of n, a(n) for n = 0..100
D. Birmajer, J. B. Gil, J. D. Gil and M. D. Weiner, Schröder Coloring and Applications, arXiv:1908.08103 [math.CO], 2019.

Cf. A153231 (x+2*x^3).

[2^n*Binomial(4*n, n)/(3*n+1): n in [0..25]]; // Vincenzo Librandi, Jul 26 2019

A100089 := proc(n)
    (3*n+1)! ;
end proc:
A060706 := proc(n)
    (4*n)!/n!/4^n ;
end proc:
A217360 := proc(n)
    8^(n)*A060706(n)/A100089(n) ;
end proc:
seq(A217360(n), n=0..20);

Table[2^n Binomial[4 n, n] / (3 n + 1), {n, 0, 20}] (* Vincenzo Librandi, Jul 26 2019 *)

D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n)- 8*(4*n-1)*(4*n-3)*(4*n-2)*a(n-1) = 0, so a(n) = 8^n*A060706(n)/A100089(n) = 2^n*A002293(n).
a(n) = [x^(3*n)](f(x)/x) where f(x) is the reversion of x - 2*x^4.
G.f.: F([1/4, 1/2, 3/4], [2/3, 4/3], 512*x/27), where F is the generalized hypergeometric function. - Stefano Spezia, Aug 18 2019
G.f. A(x) satisfies: A(x) = 1 / (1 - 2 * x * A(x)^3). - Ilya Gutkovskiy, Nov 12 2021

Offset decreased by 1 and name changed by Michael D. Weiner, Jul 25 2019

A217363 Series reversion of x - 3*x^3.

Original entry on oeis.org

1, 3, 27, 324, 4455, 66339, 1041012, 16953624, 283848543, 4855304025, 84482290035, 1490628232080, 26607713942628, 479621100042756, 8718235759397880, 159628084420459248, 2941328850997439439, 54501093415540789605, 1014898739548854163185

Offset: 1

R. J. Mathar, Oct 01 2012

Regular zeros in the reverted sequence have been left out: If y = x - 3*x^3, then x = y + 3*y^3 + 27*y^5 + 324*y^7 + 4455*y^9 + 66339*y^11 + ...

Number of lattice paths that do not go below the x-axis from (0,0) to (3n,0) using steps D(1,-1) and three types of U(1,2). - David Scambler, Jun 22 2013

Robert Israel, Table of n, a(n) for n = 1..769
R. J. Mathar, Series Expansion of Generalized Fresnel Integrals, arXiv:1211.3963 [math.CA], 2012, App. A.

Cf. A005159 (revert x-3*x^2), A153231 (revert x-2*x^3).

f:= k -> (3*k-3)!*3^(k-1)/(k-1)!/(2*k-1)!:
map(f, [$1..30]); # Robert Israel, May 07 2017

CoefficientList[Series[2/3 Sqrt[z] Sin[ArcSin[(9 Sqrt[z])/2]/3], {z, 0, 20}], z](* Benedict W. J. Irwin, Jul 12 2016 *)

D-finite with recurrence (2*n-1)*(2*n-2)*a(n) - 9*(3*n-4)*(3*n-5)*a(n-1) = 0.
a(n) = 3^(n-1)*A001764(n-1).
From Benedict W. J. Irwin, Jul 12 2016: (Start)
G.f.: (2/3)*sqrt(x)*sin(asin(9*sqrt(x)/2)/3).
E.g.f.: x*2F2(1/3,2/3;3/2,2;81*x/4). (End)
a(n) ~ 3^(4*n - 7/2)*4^(-n)*n^(-3/2)/sqrt(Pi). - Ilya Gutkovskiy, Jul 12 2016

A369510 Expansion of (1/x) * Series_Reversion( x * ((1-x)^2-x^2)^2 ).

Original entry on oeis.org

1, 4, 28, 240, 2288, 23296, 248064, 2728704, 30764800, 353633280, 4128783360, 48827351040, 583674642432, 7041154416640, 85610725769216, 1048040981594112, 12907157115568128, 159802897621319680, 1987875305403187200, 24833149969036738560, 311409431144819589120

Offset: 0

Seiichi Manyama, Jan 25 2024

nonn

a(n) also counts triangulations of a convex (2n+3)-gon whose points are colored red and blue alternatingly, and that do not have monochromatic triangles (i.e., every triangle has at least one red point and at least one blue point). - Torsten Muetze, May 08 2024

Bruce E. Sagan, Proper partitions of a polygon and k-Catalan numbers, Ars Combinatoria, 88 (2008), 109-124.

CombOS - Combinatorial Object Server, Generate k-ary trees and dissections
Bruce E. Sagan, Proper partitions of a polygon and k-Catalan numbers, arXiv:math/0407280 [math.CO], 2004.
Index entries for reversions of series

Cf. A368961, A369513.
Cf. A151374.
Cf. A153231 (colorful triangulations with an even number of points).

my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^2-x^2)^2)/x)

a(n) = sum(k=0, n\2, binomial(2*n+k+1, k)*binomial(5*n+3, n-2*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+k+1,k) * binomial(5*n+3,n-2*k).
From Torsten Muetze, May 08 2024: (Start)
a(n) = 2^n/(n+1) * binomial(3n+1,n).
a(n) = 2^n*A006013(n). (End)

cp's OEIS Frontend

A367389 Expansion of g.f. A(x) = B(x^2)/(1 - 2*x*B(x^2)) where B(x) = 1 + 2*x*B(x)^3 is the g.f. of A153231.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A349253 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - 2 * x * A(x)^2)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A228484 a(n) = 2^n*(3*n)!/(n!*(2*n)!).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A349255 G.f. A(x) satisfies A(x) = 1 / ((1 + x) * (1 - 2 * x * A(x)^2)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A364167 Expansion of g.f. A(x) satisfying A(x) = 1 + x * A(x)^3 * (1 + A(x)^3).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

Formula

A217364 a(n) = 2^n*binomial(5*n, n)/(4*n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

Extensions

A348912 G.f. A(x) satisfies A(x) = (1 + 2 * x * A(x)^3) / (1 - x).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A367389 Expansion of g.f. A(x) = B(x^2)/(1 - 2xB(x^2)) where B(x) = 1 + 2xB(x)^3 is the g.f. of A153231.

A228484 a(n) = 2^n(3n)!/(n!(2n)!).

A217364 a(n) = 2^nbinomial(5n, n)/(4*n+1).

A217360 a(n) = 2^nbinomial(4n, n)/(3*n+1).