A048990 -id:A048990 - cp's OEIS frontend

A245113 G.f. A(x) satisfies A(x)^2 = 1 + 4xA(x)^6.

1, 2, 22, 340, 6118, 120060, 2492028, 53798888, 1195684230, 27175425004, 628705751828, 14756641134872, 350529497005532, 8410852483002200, 203561027031883320, 4963404936414528720, 121810229481173225670, 3006555636255509030220, 74585744314812449403300, 1858695101618327423328312

Offset: 0

Paul D. Hanna, Jul 31 2014

Radius of convergence of g.f. A(x) is r = 1/27 where A(r) = sqrt(3/2).

			G.f.: A(x) = 1 + 2*x + 22*x^2 + 340*x^3 + 6118*x^4 + 120060*x^5 + ...
where A(x)^2 = 1 + 4*x*A(x)^6:
A(x)^2 = 1 + 4*x + 48*x^2 + 768*x^3 + 14080*x^4 + 279552*x^5 + ...
A(x)^6 = 1 + 12*x + 192*x^2 + 3520*x^3 + 69888*x^4 + 1462272*x^5 + ...
Related series:
A(x)^5 = 1 + 10*x + 150*x^2 + 2660*x^3 + 51750*x^4 + 1068012*x^5 + ...
A(x)^10 = 1 + 20*x + 400*x^2 + 8320*x^3 + 179200*x^4 + 3969024*x^5 + ...
where A(x) = sqrt(1 + 4*x^2*A(x)^10) + 2*x*A(x)^5.

Cf. A245114.

Cf. A000984, A048990, A078531, A182122, A245112.

A245113:=n->4^n*binomial((6*n-1)/2,n)/(4*n+1): seq(A245113(n), n=0..30); # Wesley Ivan Hurt, Aug 11 2015

Table[4^n*Binomial[(6 n - 1)/2, n]/(4 n + 1), {n, 0, 20}] (* Wesley Ivan Hurt, Aug 11 2015 *)

/* From A(x)^2 = 1 + 4*x*A(x)^6 : */
{a(n) = local(A=1+x);for(i=1,n,A=sqrt(1 + 4*x*A^6 +x*O(x^n)));polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

{a(n) = 4^n * binomial((6*n - 1)/2, n) / (4*n + 1)}
for(n=0,20,print1(a(n),", "))

/* From A(x) = sqrt(1 + 4*x^2*A(x)^10) + 2*x*A(x)^5 : */
{a(n) = local(A=1+x);for(i=1,n,A = sqrt(1 + 4*x^2*A^10 +x*O(x^n)) + 2*x*A^5);polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

a(n) = 4^n * binomial((6*n - 1)/2, n) / (4*n + 1).
G.f. A(x) satisfies A(x) = sqrt(1 + 4*x^2*A(x)^10) + 2*x*A(x)^5.
G.f.: A(x) = sqrt(D(4*x)), where D(x) is the g.f. of A001764. - Werner Schulte, Aug 10 2015
From Karol A. Penson, Mar 19 2024: (Start)
a(n) = 4^n*binomial(3*n+1/2,n)/(6*n+1).
G.f.: 3F2([1/6, 1/2, 5/6], [3/4, 5/4], 27*x).
G.f.: sqrt(2)*sqrt((-(sqrt(1 - 27*x) + 3*i*sqrt(3)*sqrt(x))^(1/3) + (sqrt(1 - 27*x) - 3*i*sqrt(3)*sqrt(x))^(1/3))*i)*3^(3/4)/(6*x^(1/4)), where i is the imaginary unit.
a(n) = Integral_{x=0...27} x^n*W(x), where W(x) = h1(x) + h2(x) + h3(x) and
h1(x) = 2^(2/3)*3F2([-1/12, 1/6, 5/12], [1/3, 2/3], x/27)/(4*Pi*x^(5/6));
h2(x) = -3F2([1/4, 1/2, 3/4], [2/3, 4/3], x/27)/(12*Pi*sqrt(x));
h3(x) = -2^(1/3)*3F2([7/12, 5/6, 13/12], [4/3, 5/3], x/27)/(576*Pi*x^(1/6)).
This integral representation is unique as W(x) is the solution of the Hausdorff power moment problem on x = (0, 27). Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0 and for x > 0 is monotonically decreasing to zero at x = 27. For x -> 27, W'(x) tends to -infinity. (End)
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^10). - Seiichi Manyama, Jun 20 2025
a(n) ~ 27^n / (4 * n^(3/2) * sqrt(2*Pi)). - Amiram Eldar, Sep 04 2025

A187359 Catalan trisection: A000108(3*n + 2)/2, n>=0.

Original entry on oeis.org

1, 21, 715, 29393, 1337220, 64822395, 3282060210, 171529806825, 9183676536076, 501121108325684, 27767032438524099, 1558142747453650631, 88366931393503350700, 5056959295818949067010, 291650059796498346544020, 16934386878595523443214745, 989130828878080326811887228, 58078935727891217125276922940, 3426228463922436748774829232156, 202972497563788492865321721683556

Offset: 0

Wolfdieter Lang, Mar 09 2011

See the comment under A187357 for the o.g.f.s of the general trisection of a sequence.

The sequence C(3*n+2) starts as 2, 42, 1430, 58786, 2674440, 129644790, 6564120420, 343059613650, ...

Cf. A000108, A024492, A048990, A187357 (C(3*n)), A187358 (C(3*n+1)).

Table[CatalanNumber[3*n+2]/2, {n, 0, 20}] (* Amiram Eldar, Mar 16 2022 *)

a(n) = C(3*n+2)/2, n>=0, with C(n) = A000108(n).
O.g.f.: (3 - sqrt(1 - 4*x^(1/3)) - sqrt(2)*sqrt(sqrt(1 + 4*x^(1/3) + 16*x^(2/3)) +
  (1 + 2*x^(1/3))))/(12*x).
From Ilya Gutkovskiy, Jan 21 2017: (Start)
E.g.f.: 3F3(5/6,7/6,3/2; 4/3,5/3,2; 64*x).
a(n) ~ 8^(2*n+1)/(3*sqrt(3*Pi)*n^(3/2)). (End)
Sum_{n>=0} a(n)/4^n = 1 - sqrt(3+2*sqrt(3))/3. - Amiram Eldar, Mar 16 2022
a(n) = (1/2)*Product_{1 <= i <= j <= 3*n+1} (3*i + j + 2)/(3*i + j - 1). - Peter Bala, Feb 22 2023

A235534 a(n) = binomial(6n, 2n) / (4*n + 1).

Original entry on oeis.org

1, 3, 55, 1428, 43263, 1430715, 50067108, 1822766520, 68328754959, 2619631042665, 102240109897695, 4048514844039120, 162250238001816900, 6568517413771094628, 268225186597703313816, 11034966795189838872624, 456949965738717944767791

Offset: 0

Bruno Berselli, Jan 12 2014

This is the case l=4, k=2 of binomial((l+k)*n,k*n)/((l*n+1)/gcd(k,l*n+1)), see Theorem 1.1 in Zhi-Wei Sun's paper.

First bisection of A001764.

Zhi-Wei Sun, On Divisibility Of Binomial Coefficients, Journal of the Australian Mathematical Society 93 (2012), p. 189-201.

Cf. similar sequences generated by binomial((l+k)*n,k*n)/(l*n+1), where l is divisible by all the factors of k: A000108 (l=1, k=1), A001764 (l=2, k=1), A002293 (l=3, k=1), A002294 (l=4, k=1), A002295 (l=5, k=1), A002296 (l=6, k=1), A007556 (l=7, k=1), A062994 (l=8, k=1), A059968 (l=9, k=1), A230388 (l=10, k=1), A048990 (l=2, k=2), this sequence (l=4, k=2), A235536 (l=6, k=2), A187357 (l=3, k=3), A235535 (l=6, k=3).

l:=4; k:=2; [Binomial((l+k)*n,k*n)/(l*n+1): n in [0..20]]; /* where l is divisible by all the prime factors of k */

Table[Binomial[6 n, 2 n]/(4 n + 1), {n, 0, 20}]

a(n) = A047749(4*n-2) for n>0.
From Ilya Gutkovskiy, Jun 21 2018: (Start)
G.f.: 4F3(1/6,1/3,2/3,5/6; 1/2,3/4,5/4; 729*x/16).
a(n) ~ 3^(6*n+1/2)/(sqrt(Pi)*2^(4*n+7/2)*n^(3/2)). (End)

A235535 a(n) = binomial(9n, 3n) / (6*n + 1).

Original entry on oeis.org

1, 12, 1428, 246675, 50067108, 11124755664, 2619631042665, 642312451217745, 162250238001816900, 41932353590942745504, 11034966795189838872624, 2946924270225408943665279, 796607831560617902288322405, 217550867863011281855594752680

Offset: 0

Bruno Berselli, Jan 12 2014

This is the case l=6, k=3 of binomial((l+k)*n,k*n)/((l*n+1)/gcd(k,l*n+1)), see Theorem 1.1 in Zhi-Wei Sun's paper.

Also, the sequence follows A002296 and A235536, namely binomial(7*n,n)/(6*n+1) and binomial(8*n,2*n)/(6*n+1); naturally, even binomial(10*n,4*n)/(6*n+1) is always integer.

Robert Israel, Table of n, a(n) for n = 0..403
Zhi-Wei Sun, On Divisibility Of Binomial Coefficients, Journal of the Australian Mathematical Society 93 (2012), p. 189-201.

Cf. similar sequences generated by binomial((l+k)*n,k*n)/(l*n+1), where l is divisible by all the factors of k: A000108 (l=1, k=1), A001764 (l=2, k=1), A002293 (l=3, k=1), A002294 (l=4, k=1), A002295 (l=5, k=1), A002296 (l=6, k=1), A007556 (l=7, k=1), A062994 (l=8, k=1), A059968 (l=9, k=1), A230388 (l=10, k=1), A048990 (l=2, k=2), A235534 (l=4, k=2), A235536 (l=6, k=2), A187357 (l=3, k=3), this sequence (l=6, k=3).

l:=6; k:=3; [Binomial((l+k)*n,k*n)/(l*n+1): n in [0..20]]; /* here l is divisible by all the prime factors of k */

seq(binomial(9*n,3*n)/(6*n+1), n=0..30); # Robert Israel, Feb 15 2021

Table[Binomial[9 n, 3 n]/(6 n + 1), {n, 0, 20}]

a(n) = A001764(3*n) = A047749(6*n).
From Ilya Gutkovskiy, Jun 21 2018: (Start)
G.f.: 6F5(1/9,2/9,4/9,5/9,7/9,8/9; 1/3,1/2,2/3,5/6,7/6; 19683*x/64).
a(n) ~ 3^(9*n-1)/(sqrt(Pi)*4^(3*n+1)*n^(3/2)). (End)
D-finite with recurrence 8*(6*n + 5)*(2*n + 1)*(n + 1)*(3*n + 2)*(3*n + 1)*(6*n + 7)*a(n + 1) = 3*(9*n + 8)*(9*n + 7)*(9*n + 5)*(9*n + 4)*(9*n + 2)*(9*n + 1)*a(n). - Robert Israel, Feb 15 2021

A206300 Expand the real root of y^3 - y + x in powers of x, then multiply coefficient of x^n by -4^n to get integers.

Original entry on oeis.org

-1, 2, 6, 32, 210, 1536, 12012, 98304, 831402, 7208960, 63740820, 572522496, 5209363380, 47915728896, 444799488600, 4161823309824, 39209074920090, 371626340253696, 3541117629057540

Offset: 0

Roger L. Bagula, Feb 05 2012

Also coefficients of the series S(u) for which (-sqrt(3u))*S converges to the larger of the two real roots of x^3 - 3ux + 4u for u >= 4. Specifically, S(u)=Sum_{n>=0} a(n)/(27*u)^(n/2). - Dixon J. Jones, Jun 24 2021

George E. Andrews, Number Theory, 1971, Dover Publications, New York, pp. 41-43.

Cf. A000108, A048990, A224884 (signed version).

Cf. A085614.

p[x_] = y /. Solve[y^3 - y + x == 0, y][[1]]
b = Table[-4^n*FullSimplify[ExpandAll[SeriesCoefficient[ Series[p[x], {x, 0, 30}], n]]], {n, 0, 30}]
Table[2^(2n - 1) Gamma[(3n - 1)/2]/(Gamma[(n + 1)/2]n!), {n, 0, 20}] (* Dixon J. Jones, Jun 24 2021 *)
Table[2^(2n - 1) Pochhammer[(n + 1)/2, (n-1)]/n!, {n, 0, 20}] (* Dixon J. Jones, Jun 24 2021 *)

-x/serreverse((x*sqrt(1-4*x))) \\ Thomas Baruchel, Jul 02 2018

G.f.: -(12*x)/(2*sin(arcsin(216*x^2-1)/3)+1). - Vladimir Kruchinin, Oct 30 2014
G.f.: -x/Revert((x*sqrt(1-4*x))). - Thomas Baruchel, Jul 02 2018
G.f.: - (1/x) * Revert( x*sqrt(c(4*x)) ), where c(x) =  (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108 and sqrt(c(4*x)) is the g.f. of A048990. - Peter Bala, Mar 05 2020
From Dixon J. Jones, Jun 24 2021: (Start)
a(n) = 2*A085614(n) for n>=1.
a(n) = 2^(2*n - 1) Gamma((3*n - 1)/2)/(Gamma((n + 1)/2)*n!).
a(n) = (2^(2*n - 1)*((n + 1)/2)_(n-1))/n!, where (x)_k is the Pochhammer symbol. (End)
a(n) ~ 2^(n-1/2) * 3^(3*n/2-1) / (sqrt(Pi) * n^(3/2)). - Amiram Eldar, Sep 01 2025

Edited by N. J. A. Sloane, Feb 09 2012

A235536 a(n) = binomial(8n, 2n) / (6*n + 1).

Original entry on oeis.org

1, 4, 140, 7084, 420732, 27343888, 1882933364, 134993766600, 9969937491420, 753310723010608, 57956002331347120, 4524678117939182220, 357557785658996609700, 28545588568201512137904, 2298872717007844035521848, 186533392975795702301759056

Offset: 0

Bruno Berselli, Jan 12 2014

This is the case l=6, k=2 of binomial((l+k)*n,k*n)/((l*n+1)/gcd(k,l*n+1)), see Theorem 1.1 in Zhi-Wei Sun's paper.
First bisection of A002293.
Also, the sequence is between A002296 and A235535.

Zhi-Wei Sun, On Divisibility Of Binomial Coefficients, Journal of the Australian Mathematical Society 93 (2012), p. 189-201.

Cf. similar sequences generated by binomial((l+k)*n,k*n)/(l*n+1), where l is divisible by all the factors of k: A000108 (l=1, k=1), A001764 (l=2, k=1), A002293 (l=3, k=1), A002294 (l=4, k=1), A002295 (l=5, k=1), A002296 (l=6, k=1), A007556 (l=7, k=1), A062994 (l=8, k=1), A059968 (l=9, k=1), A230388 (l=10, k=1), A048990 (l=2, k=2), A235534 (l=4, k=2), this sequence (l=6, k=2), A187357 (l=3, k=3), A235535 (l=6, k=3).

l:=6; k:=2; [Binomial((l+k)*n,k*n)/(l*n+1): n in [0..20]]; /* where l is divisible by all the prime factors of k */

Table[Binomial[8 n, 2 n]/(6 n + 1), {n, 0, 20}]

a(n) = A124753(6*n).
From Ilya Gutkovskiy, Jun 21 2018: (Start)
G.f.: 6F5(1/8,1/4,3/8,5/8,3/4,7/8; 1/3,1/2,2/3,5/6,7/6; 65536*x/729).
a(n) ~ 2^(16*n-1)/(sqrt(Pi)*3^(6*n+3/2)*n^(3/2)). (End)

A276483 Decimal expansion of Sum_{k>=0} (2k+1)/binomial(4k,2*k).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Sep 05 2016

			1.57976837955402077524299785912344486...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's MathWorld, Catalan Number

Cf. A000108, A001622, A048990, A121839, A268813.

SetDefaultRealField(RealField(100)); R:=RealField(); 2*Pi(R)/(9*Sqrt(3)) - 4*(3*Sqrt(5)*Log((1+Sqrt(5))/2) - 40)/125; // G. C. Greubel, Nov 04 2018

RealDigits[2 (Pi/(9 Sqrt[3])) - 4 ((3 Sqrt[5] Log[GoldenRatio] - 40)/125), 10, 120][[1]]
RealDigits[HypergeometricPFQ[{1, 1, 3/2}, {1/4, 3/4}, 1/16], 10, 120][[1]]

suminf(k=0, 1/(binomial(4*k,2*k)/(2*k+1))) \\ Michel Marcus, Sep 06 2016

default(realprecision, 100); 2*Pi/(9*sqrt(3)) - 4*(3*sqrt(5)*log((1+sqrt(5))/2) - 40)/125 \\ G. C. Greubel, Nov 04 2018

Equals 2*Pi/(9*sqrt(3)) - 4*(3*sqrt(5)*log(phi) - 40)/125, where phi is the golden ratio (A001622).
Equals Sum_{k>=0} 1/Catalan number(2k).
Equals Sum_{k>=0} 1/A000108(2k).
Equals Sum_{k>=0} 1/A048990(k).

A098465 Expansion of (sqrt(1+3x)-sqrt(1-5x))/(4xsqrt(1-x)).

Original entry on oeis.org

1, 1, 3, 7, 27, 91, 373, 1457, 6163, 25795, 111897, 486421, 2153429, 9584901, 43121211, 195082479, 888861555, 4069956979, 18732710281, 86579713685, 401776434017, 1870946532705, 8740907398527, 40956105551603

Offset: 0

Paul Barry, Sep 09 2004

Binomial transform of A048990 (with interpolated zeros).

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

Cf. A000108.

CoefficientList[Series[(Sqrt[1+3*x]-Sqrt[1-5*x])/(4*x*Sqrt[1-x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

x='x+O('x^66); Vec((sqrt(1+3*x)-sqrt(1-5*x))/(4*x*sqrt(1-x))) \\ Joerg Arndt, May 11 2013

a(n) = sum{k=0..n} binomial(n,k) * C(k) * (1+(-1)^k)/2.
Recurrence: n*(n+1)*a(n) = 2*n*(2*n-1)*a(n-1) + 2*(5*n^2-10*n+3)*a(n-2) - 14*(n-2)*(2*n-3)*a(n-3) + 15*(n-3)*(n-2)*a(n-4). - Vaclav Kotesovec, Oct 24 2012
a(n) ~ 5^(n+3/2)/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012

A098469 A sequence related to the even-indexed Catalan numbers.

Original entry on oeis.org

1, 2, 6, 20, 78, 332, 1516, 7240, 35734, 180620, 929940, 4858328, 25687052, 137177016, 738819672, 4008435984, 21886788582, 120178329740, 663179894788, 3675923244856, 20456707469540, 114254175491304, 640223315385576

Offset: 0

Paul Barry, Sep 09 2004, corrected Mar 31 2007

Binomial transform of A098465. Second binomial transform of (1,0,2,0,14,0,132,0,1430,...) (set odd-indexed Catalan numbers to zero).

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

Cf. A048990.

CoefficientList[Series[(Sqrt[1+2*x]-Sqrt[1-6*x])/(4*x*Sqrt[1-2*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

x='x+O('x^66); Vec((sqrt(1+2*x)-sqrt(1-6*x))/(4*x*sqrt(1-2*x))) \\ Joerg Arndt, May 11 2013

G.f.: (sqrt(1+2*x) - sqrt(1-6*x))/(4*x*sqrt(1-2*x)).
a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*C(k)*2^(n-2k).
a(n) = Sum_{k=0..n} C(n,k)*2^(n-k)*C(k)*(1-(-1)^k)/2.
Recurrence: n*(n+1)*a(n) = 4*n*(2*n-1)*a(n-1) - 4*(2*n^2 - 4*n + 3)*a(n-2) - 16*(n-2)*(2*n-3)*a(n-3) + 48*(n-3)*(n-2)*a(n-4). - Vaclav Kotesovec, Oct 24 2012
a(n) ~ 3*6^(n+1/2)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012

A133602 The matrix-vector product A133080 * A000108.

Original entry on oeis.org

1, 2, 2, 7, 14, 56, 132, 561, 1430, 6292, 16796, 75582, 208012, 950912, 2674440, 12369285, 35357670, 165002460, 477638700, 2244901890, 6564120420, 31030387440, 91482563640, 434542177290, 1289904147324, 6151850548776

Offset: 0

Gary W. Adamson, Sep 18 2007

A133603 is a companion sequence.

			a(4) = C(4) = 14.
a(5) = 56 = C(5) + C(4) = 42 + 14.

Cf. A097806, A133080, A000108, A133603.

from sympy import catalan
def a005807(n): return catalan(n) + catalan(n + 1)
def a048990(n): return catalan(2*n)
l=[1, 2]
for n in range(2, 31): l+=[a048990(n//2) if n%2==0 else a005807(n - 1)]
print(l) # Indranil Ghosh, Jul 15 2017

A133080 * A000108, where A133080 = an infinite lower triangular matrix and A000108 = the Catalan sequence as a vector.
a(2n) = A048990(n).
a(2n+1) = A005807(2n).
Conjecture: n*(n-1)*(n-3)*(3*n-4)*a(n) -8*(n-1)*(2*n-5)*a(n-1) -4*(n-2)*(3*n-1)*(2*n-5)*(2*n-7)*a(n-2)=0. - R. J. Mathar, Jun 20 2015

cp's OEIS Frontend

A245113 G.f. A(x) satisfies A(x)^2 = 1 + 4*x*A(x)^6.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A187359 Catalan trisection: A000108(3*n + 2)/2, n>=0.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A235534 a(n) = binomial(6*n, 2*n) / (4*n + 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A235535 a(n) = binomial(9*n, 3*n) / (6*n + 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A206300 Expand the real root of y^3 - y + x in powers of x, then multiply coefficient of x^n by -4^n to get integers.

Views

Author

Keywords

Comments

References

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A235536 a(n) = binomial(8*n, 2*n) / (6*n + 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A276483 Decimal expansion of Sum_{k>=0} (2*k+1)/binomial(4*k,2*k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

A245113 G.f. A(x) satisfies A(x)^2 = 1 + 4xA(x)^6.

A235534 a(n) = binomial(6n, 2n) / (4*n + 1).

A235535 a(n) = binomial(9n, 3n) / (6*n + 1).

A235536 a(n) = binomial(8n, 2n) / (6*n + 1).

A276483 Decimal expansion of Sum_{k>=0} (2k+1)/binomial(4k,2*k).

A098465 Expansion of (sqrt(1+3x)-sqrt(1-5x))/(4xsqrt(1-x)).