A182122 -id:A182122 - cp's OEIS frontend

A048990 Catalan numbers with even index (A000108(2n), n >= 0): a(n) = binomial(4n, 2n)/(2n+1).

1, 2, 14, 132, 1430, 16796, 208012, 2674440, 35357670, 477638700, 6564120420, 91482563640, 1289904147324, 18367353072152, 263747951750360, 3814986502092304, 55534064877048198, 812944042149730764, 11959798385860453492, 176733862787006701400

Offset: 0

Wolfdieter Lang

With interpolated zeros, this is C(n)*(1+(-1)^n)/2 with g.f. given by 2/(sqrt(1+4x) + sqrt(1-4x)). - Paul Barry, Sep 09 2004
Self-convolution of a(n)/4^n gives Catalan numbers (A000108). - Vladimir Reshetnikov, Oct 10 2016
a(n) is the number of grand Dyck paths from (0,0) to (4n,0) that avoid vertices (2k,0) for all odd k > 0. - Alexander Burstein, May 11 2021
a(n) is the number of lattice paths from (0,0) to (2n,2n) with steps (1,0) and (0,1) that avoid the points (1,1), (3,3), (5,5), ..., (2n-1,2n-1).  This is Example 2.5 of the Shapiro reference. - Lucas A. Brown, Jul 24 2025

			sqrt(2*x^-1*(1-sqrt(1-x))) = 1 + (1/8)*x + (7/128)*x^2 + (33/1024)*x^3 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.
Gi-Sang Cheon, S.-T. Jin, and L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Vol. 491 (2016), pp. 123-137.
Greg Markowsky, A method for deriving hypergeometric and related identities from the H 2 Hardy norm of conformal maps, Integral Transforms and Special Functions, Vol. 24, No. 4 (2013), pp. 302-313; arXiv preprint, arXiv:1205.2458 [math.CV], 2012.
Khodabakhsh Hessami Pilehrood and Tatiana Hessami Pilehrood, Jacobi Polynomials and Congruences Involving Some Higher-Order Catalan Numbers and Binomial Coefficients, J. Int. Seq., Vol. 18 (2015), Article 15.11.7.
Louis Shapiro, Random walks with absorbing points, Discrete Applied Mathematics, Vol. 39 (1992), pp. 57-67.

Cf. A000007, A000108, A024492, A065097, A099250, A187357, A187358, A187359, A228411, A276483.

Cf. A000984, A078531, A182122, A245112, A245113.

a[n_] := CatalanNumber[2n]; Array[a, 18, 0] (* Or *)
CoefficientList[ Series[ Sqrt[2]/Sqrt[1 + Sqrt[1 - 16 x]], {x, 0, 17}], x] (* Robert G. Wilson v *)
CatalanNumber[Range[0,40,2]] (* Harvey P. Dale, Mar 19 2015 *)

combinat::dyckWords::count(2*n) $ n = 0..28 // Zerinvary Lajos, Apr 14 2007

/* G.f.: A(x) = exp( x*A(x)^4 + Integral(A(x)^4 dx) ): */
{a(n)=local(A=1+x); for(i=1, n, A=exp(x*A^4 + intformal(A^4 +x*O(x^n)))); polcoeff(A, n)} \\ Paul D. Hanna, Nov 09 2013
for(n=0, 30, print1(a(n), ", "))

A048990 = lambda n: hypergeometric([1-2*n,-2*n],[2],1)
[Integer(A048990(n).n()) for n in range(20)] # Peter Luschny, Sep 22 2014

a(n) = 2 * A065097(n) - A000007(n).
G.f.: A(x) = sqrt((1/8)*x^(-1)*(1-sqrt(1-16*x))).
G.f.: 2F1( (1/4, 3/4); (3/2))(16*x). - Olivier Gérard Feb 17 2011
D-finite with recurrence n*(2*n+1)*a(n) - 2*(4*n-1)*(4*n-3)*a(n-1) = 0. - R. J. Mathar, Nov 30 2012
E.g.f: 2F2(1/4, 3/4; 1, 3/2; 16*x). - Vladimir Reshetnikov, Apr 24 2013
G.f. A(x) satisfies: A(x) = exp( x*A(x)^4 + Integral(A(x)^4 dx) ). - Paul D. Hanna, Nov 09 2013
G.f. A(x) satisfies: A(x) = sqrt(1 + 4*x*A(x)^4). - Paul D. Hanna, Nov 09 2013
a(n) = hypergeom([1-2*n,-2*n],[2],1). - Peter Luschny, Sep 22 2014
a(n) ~ 2^(4*n-3/2)/(sqrt(Pi)*n^(3/2)). - Ilya Gutkovskiy, Oct 10 2016
From Peter Bala, Feb 27 2020: (Start)
a(n) = (4^n)*binomial(2*n + 1/2, n)/(4*n + 1).
O.g.f.: A(x) = sqrt(c(4*x)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers. Cf. A228411. (End)
Sum_{n>=0} 1/a(n) = A276483. - Amiram Eldar, Nov 18 2020
Sum_{n>=0} a(n)/4^n = sqrt(2). - Amiram Eldar, Mar 16 2022
From Peter Bala, Feb 22 2023: (Start)
a(n) = (1/2^(2*n-1)) * Product_{1 <= i <= j <= 2*n-1} (i + j + 2)/(i + j - 1) for n >= 1.
a(n) = Product_{1 <= i <= j <= 2*n-1} (3*i + j + 2)/(3*i + j - 1). Cf. A024492. (End)
a(n) = Sum_{k = 0..2*n-1} (-1)^k * 4^(2*n-k-1)*binomial(2*n-1, k)*Catalan(k+1). - Peter Bala, Apr 29 2024
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^6). - Seiichi Manyama, Jun 20 2025

A385205 G.f. A(x) satisfies A(x) = ( 1 + 25xA(x)^4 )^(1/5).

Original entry on oeis.org

1, 5, 50, 500, 4375, 27500, 0, -3562500, -70078125, -876562500, -6926562500, 0, 1189169921875, 25690820312500, 346441406250000, 2911880859375000, 0, -550017993164062500, -12339622131347656250, -171953389892578125000, -1487552714691162109375, 0

Offset: 0

Seiichi Manyama, Jun 21 2025

sign

Cf. A078534, A299958, A385203, A385204.

Cf. A182122, A376636, A385207.

a(n) = 25^n*binomial(4*n/5+1/5, n)/(4*n+1);

a(n) = 25^n * binomial(4*n/5+1/5,n)/(4*n+1).
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^3).
G.f.: ( (1/x) * Series_Reversion(x/(1+25*x)^(4/5)) )^(1/4).
a(5*n+1) = 0 for n > 0.
G.f.: 1/B(x), where B(x) is the g.f. of A299958.

A224884 Expansion of x / Series_Reversion(xsqrt(1 + 4x)).

Original entry on oeis.org

1, 2, -6, 32, -210, 1536, -12012, 98304, -831402, 7208960, -63740820, 572522496, -5209363380, 47915728896, -444799488600, 4161823309824, -39209074920090, 371626340253696, -3541117629057540, 33902753847705600, -325969196485349340, 3146175557067079680, -30471769822097981160

Offset: 0

Paul D. Hanna, Aug 21 2013

sign

Signed version of A206300. - Peter Bala, Mar 05 2020

			G.f.: A(x) = 1 + 2*x - 6*x^2 + 32*x^3 - 210*x^4 + 1536*x^5 - 12012*x^6 + ..
The coefficients in the powers A(x)^n of the g.f. begin:
n= 1: [1,  2,  -6,   32,  -210,  1536,-12012,  98304, -831402, ...];
n= 2: [1,  4,  -8,   40,  -256,  1848,-14336, 116688, -983040, ...];
n= 3: [1,  6,  -6,   32,  -210,  1536,-12012,  98304, -831402, ...];
n= 4: [1,  8,   0,   16,  -128,  1008, -8192,  68640, -589824, ...];
n= 5: [1, 10,  10,    0,   -50,   512, -4620,  40960, -364650, ...];
n= 6: [1, 12,  24,   -8,     0,   168, -2048,  20592, -196608, ...];
n= 7: [1, 14,  42,    0,    14,     0,  -588,   8192,  -90090, ...];
n= 8: [1, 16,  64,   32,     0,   -32,     0,   2112,  -32768, ...];
n= 9: [1, 18,  90,   96,   -18,     0,    84,      0,   -7722, ...];
n=10: [1, 20, 120,  200,     0,    24,     0,   -240,       0, ...];
n=11: [1, 22, 154,  352,   110,     0,   -44,      0,     726, ...];
n=12: [1, 24, 192,  560,   384,   -48,     0,     96,       0, ...];
n=13: [1, 26, 234,  832,   910,     0,    52,      0,    -234, ...];
n=14: [1, 28, 280, 1176,  1792,   392,     0,    -80,       0, ...];
n=15: [1, 30, 330, 1600,  3150,  1536,  -140,      0,     150, ...];
n=16: [1, 32, 384, 2112,  5120,  4032,     0,    128,       0, ...];
n=17: [1, 34, 442, 2720,  7854,  8704,  1428,      0,    -170, ...];
n=18: [1, 36, 504, 3432, 11520, 16632,  6144,   -432,       0, ...];
n=19: [1, 38, 570, 4256, 16302, 29184, 17556,      0,     342, ...];
n=20: [1, 40, 640, 5200, 22400, 48048, 40960,   5280,       0, ...]; ...
which illustrates the property [x^n] A(x)^(n+2*k) = 0 for k=1..n-1:
[x^2] A(x)^4 = 0;
[x^3] A(x)^5 = 0, [x^3] A(x)^7 = 0;
[x^4] A(x)^6 = 0, [x^4] A(x)^8 = 0, [x^4] A(x)^10 = 0; ...
[x^5] A(x)^7 = 0, [x^5] A(x)^9 = 0, [x^5] A(x)^11 = 0, [x^5] A(x)^13 = 0; ...
Related series:
sqrt(1+4*x) = 1 + 2*x - 2*x^2 + 4*x^3 - 10*x^4 + 28*x^5 - 84*x^6 + 264*x^7 - 858*x^8 + ... + (-1)^(n-1)*2*A000108(n-1)*x^n + ...

G. C. Greubel, Table of n, a(n) for n = 0..980

Cf. A000108, A206300.

Cf. A000984, A048990, A078531, A182122, A245112, A245113.

CoefficientList[Series[x/InverseSeries[Series[x*Sqrt[1+4*x],{x,0,20}],x],{x,0,20}],x] (* Vaclav Kotesovec, Aug 22 2013 *)

{a(n)=polcoeff(x/serreverse(x*sqrt(1+4*x +x^2*O(x^n))),n)}
for(n=0,25,print1(a(n),", "))

G.f. A(x) satisfies:
(1) A(x) = A(x)^3 - 4*x.
(2) A(x) = sqrt(1 + 4*x/A(x)).
(3) A(x*sqrt(1+4*x)) = sqrt(1+4*x).
(4) [x^n] A(x)^(n+2*k) = 0 for k=1..n-1, for n >= 2.
From Vaclav Kotesovec, Aug 22 2013: (Start)
a(n) = (-1)^(n+1) * 3^(3*n/2-1) * 4^(n-1) * GAMMA(n/2 - 1/6) * GAMMA(n/2 + 1/6)/(Pi*n!).
|a(n)| ~ 6^(n-1)*3^(n/2)/(sqrt(Pi/2)*n^(3/2)).
D-finite with recurrence: (n-1)*n*a(n) = 12*(3*n-7)*(3*n-5)*a(n-2). (End)
G.f.: (2/sqrt(3))*cosh(1/3*arccosh(sqrt(108)*x)). - Vladimir Kruchinin, Oct 11 2022
G.f. A(x) satisfies A(x) = 1/A(-x/A(x)^4). - Seiichi Manyama, Jun 20 2025

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

Original entry on oeis.org

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

A385207 G.f. A(x) satisfies A(x) = ( 1 + 49xA(x)^6 )^(1/7).

Original entry on oeis.org

1, 7, 147, 3430, 79233, 1714314, 32471124, 450360372, 0, -313409171166, -15459345780879, -537166232508360, -15185812043764453, -348420909370148580, -5588125164812112720, 0, 4783756561471246040577, 254794190560328322173970, 9445124186699596552669050

Offset: 0

Seiichi Manyama, Jun 21 2025

sign

Cf. A385206, A385208.

Cf. A182122, A376636, A385205.

a(n) = 49^n*binomial(6*n/7+1/7, n)/(6*n+1);

a(n) = 49^n * binomial(6*n/7+1/7,n)/(6*n+1).
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^5).
G.f.: ( (1/x) * Series_Reversion(x/(1+49*x)^(6/7)) )^(1/6).
a(7*n+1) = 0 for n > 0.
G.f.: 1/B(-x), where B(x) is the g.f. of A385206.

A385206 G.f. A(x) satisfies A(x) = ( 1 + 49xA(x) )^(1/7).

Original entry on oeis.org

1, 7, -98, 1715, -28812, 369754, 0, -234003861, 11187831655, -379208609780, 10505577339166, -237021026782414, 3747904201751920, 0, -3136632447485449416, 165539296779239527515, -6087083256734433868530, 180571542422445599417377, -4318405727843353425012650

Offset: 0

Seiichi Manyama, Jun 21 2025

sign

Index entries for reversions of series

Cf. A385207, A385208.

Cf. A182122.

a(n) = 49^n*binomial(n/7+1/7, n)/(n+1);

my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+49*x)^(1/7))/x)

a(n) = 49^n * binomial(n/7+1/7,n)/(n+1).
G.f. A(x) satisfies A(x) = 1/A(-x/A(x)^5).
G.f.: (1/x) * Series_Reversion(x/(1+49*x)^(1/7)).
a(7*n+6) = 0 for n >= 0.

A261196 Expansion of sqrt(8x + sqrt(1 + 64x^2)).

Original entry on oeis.org

1, 4, 8, -32, -160, 896, 5376, -33792, -219648, 1464320, 9957376, -68796416, -481574912, 3408068608, 24343347200, -175272099840, -1270722723840, 9268801044480, 67971207659520, -500840477491200, -3706219533434880, 27531916534087680, 205237923254108160

Offset: 0

Werner Schulte, Aug 11 2015

Signs are important to distinguish this from (for example) A098579.

Sqrt(A(x)) = 1 + 2*x + 2*x^2 - 20*x^3 - 42*x^4 + 572*x^5 ... defines another (new) integer sequence.

			A(x) = 1 + 4*x + 8*x^2 - 32*x^3 - 160*x^4 + 896*x^5 + 5376*x^6 ...
A(x)^2 = 1 + 8*x + 32*x^2 + 0*x^3 - 512*x^4 + 0*x^5 +16384*x^6 ...
A(x)^4 = 1 + 16*x + 128*x^2 + 512*x^3 + 0*x^4 -8192*x^5 + 0*x^6 ...

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

Cf. A000108, A002420, A098579, A182122.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Sqrt(8*x+Sqrt(1+64*x^2)))); // G. C. Greubel, Aug 12 2018

CoefficientList[Series[Sqrt[8 x + Sqrt[1 + 64 x^2]], {x, 0, 45}], x] (* Vincenzo Librandi, Aug 12 2015 *)

Vec(sqrt(8*x + sqrt(1 + 64*x^2))) \\ M. F. Hasler, Aug 14 2015

G.f. A(x) satisfies: A(x)^4 = 1 + 16*x*A(x)^2.
A(x) = sqrt(G(4*x)), where G(x) is the g.f. of A182122.
A(x) * A(-x) = 1.
A(x) = sqrt(1 + 8*x + 32*x^2*C(-16*x^2)), where C(x) is the g.f. of A000108.
a(n) = A002420(n)*2^n*(-1)^(n*(n+1)/2). - M. F. Hasler, Aug 14 2015
Conjecture D-finite with recurrence: n*(n-1)*a(n) +(n-1)*(n-2)*a(n-1) +16*(2*n-3)*(2*n-5)*a(n-2) +16*(2*n-5)*(2*n-7)*a(n-3)=0. - R. J. Mathar, Jun 07 2016

cp's OEIS Frontend

A048990 Catalan numbers with even index (A000108(2*n), n >= 0): a(n) = binomial(4*n, 2*n)/(2*n+1).

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A385205 G.f. A(x) satisfies A(x) = ( 1 + 25*x*A(x)^4 )^(1/5).

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A224884 Expansion of x / Series_Reversion(x*sqrt(1 + 4*x)).

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

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A385207 G.f. A(x) satisfies A(x) = ( 1 + 49*x*A(x)^6 )^(1/7).

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A385206 G.f. A(x) satisfies A(x) = ( 1 + 49*x*A(x) )^(1/7).

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A261196 Expansion of sqrt(8*x + sqrt(1 + 64*x^2)).

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A048990 Catalan numbers with even index (A000108(2n), n >= 0): a(n) = binomial(4n, 2n)/(2n+1).

A385205 G.f. A(x) satisfies A(x) = ( 1 + 25xA(x)^4 )^(1/5).

A224884 Expansion of x / Series_Reversion(xsqrt(1 + 4x)).

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

A385207 G.f. A(x) satisfies A(x) = ( 1 + 49xA(x)^6 )^(1/7).

A385206 G.f. A(x) satisfies A(x) = ( 1 + 49xA(x) )^(1/7).

A261196 Expansion of sqrt(8x + sqrt(1 + 64x^2)).