A078531 -id:A078531 - cp's OEIS frontend

A007297 Number of connected graphs on n labeled nodes on a circle with straight-line edges that don't cross.

1, 1, 4, 23, 156, 1162, 9192, 75819, 644908, 5616182, 49826712, 448771622, 4092553752, 37714212564, 350658882768, 3285490743987, 30989950019532, 294031964658430, 2804331954047160, 26870823304476690, 258548658860327880

Offset: 1

N. J. A. Sloane, Mira Bernstein

Apart from the initial 1, reversion of g.f. for A162395 (squares with signs): see A263843.

			G.f. = x*(1 + x + 4*x^2 + 23*x^3 + 156*x^4 + 1162*x^5 + 9192*x^6 + 75819*x^7 + ...).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..999 (first 101 terms from T. D. Noe)
M. Aguiar and J.-L. Loday, Quadri-algebras, J. Pure Appl. Algebra, 191 (2004), 205-221.
R. Bacher, On generating series of complementary plane trees arXiv:math/0409050 [math.CO], 2004.
P. Barry and A. Hennessy, Four-term Recurrences, Orthogonal Polynomials and Riordan Arrays, Journal of Integer Sequences, 2012, article 12.4.2. - From _N. J. A. Sloane_, Sep 21 2012
F. R. Bernhart & N. J. A. Sloane, Emails, April-May 1994
F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).
E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004; J. Num. Theory 117 (2006), 191-215.
C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358 (column sums in Table 2).
Michael Drmota, Anna de Mier, Marc Noy, Extremal statistics on non-crossing configurations, Discrete Math. 327 (2014), 103--117. MR3192420. See Eq. (3). - N. J. A. Sloane, May 18 2014
Guillermo Esteban, Clemens Huemer, Rodrigo I. Silveira, New production matrices for geometric graphs, arXiv:2003.00524 [math.CO], 2020.
Sen-Peng Eu, Shu-Chung Liu and Yeong-Nan Yeh, On the congruences of some combinatorial numbers, Stud. Appl. Math. vol. 116 (2006) pp. 135-144.
P. Flajolet and M. Noy, Analytic Combinatorics of Non-crossing Configurations
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 486.
Loïc Foissy, Free quadri-algebras and dual quadri-algebras, arXiv:1504.06056 [math.CO], 2015.
I. M. Gessel, A short proof of the Deutsch-Sagan congruence for connected non crossing graphs, arXiv preprint arXiv:1403.7656 [math.CO], 2014.
Marco Kuhlmann, Tabulation of Noncrossing Acyclic Digraphs, arXiv:1504.04993 [cs.DS], 2015.
M. Kuhlmann, P. Jonsson, Parsing to Noncrossing Dependency Graphs, Transactions of the Association for Computational Linguistics, vol. 3, pp. 559-570, 2015.
Sara Madariaga, Gröbner-Shirshov bases for the non-symmetric operads of dendriform algebras and quadri-algebras, arXiv:1304.5184 [math.RA], 2013.
B. Vallette, Manin products, Koszul duality, Loday algebras and Deligne conjecture, arXiv:math/0609002 [math.QA], 2006-2007; J. Reine Angew. Math. 620 (2008), 105-164.
Gus Wiseman, The a(5) = 156 connected non-crossing graphs.
Gus Wiseman, Constructive Mathematica program for A007297.
Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.
Index entries for reversions of series

Cf. A162395, A000290. 4th row of A107111. Row sums of A089434.
See A263843 for a variant.
Cf. A000108 (non-crossing set partitions), A001006, A001187, A054726 (non-crossing graphs), A054921, A099947, A194560, A293510, A323818, A324167, A324169, A324173.

A007297:=proc(n) if n = 1 then 1 else add(binomial(3*n - 3, n + j)*binomial(j - 1, j - n + 1), j = n - 1 .. 2*n - 3)/(n - 1); fi; end;

CoefficientList[ InverseSeries[ Series[(x-x^2)/(1+x)^3, {x, 0, 20}], x], x] // Rest (* From Jean-François Alcover, May 19 2011, after PARI prog. *)
Table[Binomial[3n, 2n+1] Hypergeometric2F1[1-n, n, 2n+2, -1]/n, {n, 1, 20}] (* Vladimir Reshetnikov, Oct 25 2015 *)

a(n)=if(n<0,0,polcoeff(serreverse((x-x^2)/(1+x)^3+O(x^(n+2))),n+1)) /* Ralf Stephan */

Apart from initial term, g.f. is the series reversion of (x-x^2)/(1+x)^3 (A162395). See A263843. - Vladimir Kruchinin, Feb 08 2013
G.f.: (g-z)/z, where g=-1/3+(2/3)*sqrt(1+9z)*sin((1/3)*arcsin((2+27z+54z^2)/2/(1+9*z)^(3/2))). - Emeric Deutsch, Dec 02 2002
a(n) = (1/n)*Sum_{k=0..n} binomial(3n, n-k-1)*binomial(n+k-1, k). - Paul Barry, May 11 2005
a(n) = 4^(n-1)*(Gamma(3*n/2-1)/Gamma(n/2+1)/Gamma(n) -Gamma((3*n-1)/2)/ Gamma( (n+1)/2)/Gamma(n+1)). - Mark van Hoeij, Aug 27 2005, adapted to offset Feb 21 2020 by R. J. Mathar
a(n) = 4^n * binomial(3*n/2, n/2) / (9*n-6) - 4^(n-1) * binomial(3*(n-1)/2, (n-1)/2 ) / n. - Mark van Hoeij, Aug 27 2005, adapted to offset Feb 21 2020 by R. J. Mathar
D-finite with recurrence: n*(n-1)*(3*n-4)*a(n) +36*(n-1)*a(n-1) -12*(3*n-8)*(3*n-1)*(3*n-7)*a(n-2)=0. - Mark van Hoeij, Aug 27 2005, adapted to offset Feb 21 2020 by R. J. Mathar
a(n) = (1/n)*Sum_{k=0..n} C(3n, k)*C(2n-k-2, n-1). - Paul Barry, Sep 27 2005
a(n) ~ (2-sqrt(3)) * 6^n * 3^(n/2) / (sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 17 2014
a(n) = binomial(3*n,2*n+1)*hypergeom([1-n,n], [2*n+2], -1)/n. - Vladimir Reshetnikov, Oct 25 2015
a(n) = 2*A078531(n) - A085614(n+1). - Vladimir Reshetnikov, Apr 24 2016

Better description from Philippe Flajolet, Apr 20 2000
More terms from James Sellers, Aug 21 2000
Definition revised and initial a(1)=1 added by N. J. A. Sloane, Nov 05 2015 at the suggestion of Axel Boldt. Some of the formulas may now need to be adjusted slightly.

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

Original entry on oeis.org

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

A091527 a(n) = ((3n)!/n!^2)(Gamma(1+n/2)/Gamma(1+3n/2)).

Original entry on oeis.org

1, 4, 30, 256, 2310, 21504, 204204, 1966080, 19122246, 187432960, 1848483780, 18320719872, 182327718300, 1820797698048, 18236779032600, 183120225632256, 1842826521244230, 18581317012684800, 187679234340049620, 1898554215471513600, 19232182592635611060

Offset: 0

Michael Somos, Jan 18 2004

Sequence terms are given by [x^n] ( (1 + x)^(k+2)/(1 - x)^k )^n for k = 1. See the crossreferences for related sequences obtained from other values of k. - Peter Bala, Sep 29 2015

Let a > b be nonnegative integers. Then the ratio of factorials ((2*a + 1)*n)!*((b + 1/2)*n)!/(((a + 1/2)*n)!*((2*b + 1)*n)!*((a - b)*n)!) is an integer for n >= 0. This is the case a = 1, b = 0. - Peter Bala, Aug 28 2016

R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

Michael De Vlieger, Table of n, a(n) for n = 0..985
Peter Bala, Some integer ratios of factorials
Paul Barry, On the Central Coefficients of Bell Matrices, J. Int. Seq. 14 (2011) # 11.4.3.
Karl Dilcher, Armin Straub, and Christophe Vignat, Identities for Bernoulli polynomials related to multiple Tornheim zeta functions, arXiv:1903.11759 [math.NT], 2019. See p. 11.
I. M. Gessel, A short proof of the Deutsch-Sagan congruence for connected non crossing graphs, arXiv preprint arXiv:1403.7656 [math.CO], 2014.
Bernhard Heim and Markus Neuhauser, Asymptotic Distribution of the Zeros of recursively defined Non-Orthogonal Polynomials, arXiv:2107.05013 [math.CA], 2021.
W. Mlotkowski and K. A. Penson, Probability distributions with binomial moments, arXiv preprint arXiv:1309.0595 [math.PR], 2013.

Cf. A061162(n) = a(2n), A007297, A000984 (k = 0), A001448 (k = 2), A262732 (k = 3), A211419 (k = 4), A262733 (k = 5), A211421 (k = 6), A276098, A276099.

a := n -> 4^n * `if`(n<2, 1, (2*(n+1)*binomial((3*n-1)/2, n + 1))/(n-1)):
seq(a(n), n=0..18); # Peter Luschny, Feb 03 2020

Table[((3 n)!/n!^2) Gamma[1 + n/2]/Gamma[1 + 3 n/2], {n, 0, 18}] (* Michael De Vlieger, Oct 02 2015 *)
Table[4^n Sum[Binomial[k - 1 + (n - 1)/2, k], {k, 0, n}], {n, 0, 18}] (* Michael De Vlieger, Aug 28 2016 *)

B(x):=(-1/3+(2/3)*sqrt(1+9*x)*sin((1/3)*asin((2+27*x+54*x^2)/2/(1+9*x)^(3/2))))/x-1;
taylor(x*diff(B(x),x)/B(x),x,0,10); /* Vladimir Kruchinin, Oct 02 2015 */

a(n)=4^n*sum(i=0,n,binomial(i-1+(n-1)/2,i))

vector(30, n, sum(k=0, n, binomial(3*n-3, k)*binomial(2*n-k-3, n-k-1))) \\ Altug Alkan, Oct 04 2015

from math import factorial
from sympy import factorial2
def A091527(n): return int((factorial(3*n)*factorial2(n)<Chai Wah Wu, Aug 10 2023

D-finite with recurrence n*(n - 1)*a(n) = 12*(3*n - 1)*(3*n - 5)*a(n-2).
From Peter Bala, Sep 29 2015: (Start)
a(n) = Sum_{i = 0..n} binomial(3*n,i) * binomial(2*n-i-1,n-i).
a(n) = [x^n] ( (1 + x)^3/(1 - x) )^n.
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 4*x + 23*x^2 + 156*x^3 + 1162*x^4 + 9192*x^5 + ... is the o.g.f. for A007297 (but with an offset of 0). (End)
a(n) = (n+1)*A078531(n). [Barry, JIS (2011)]
G.f.: x*B'(x)/B(x), where x*B(x)+1 is g.f. of A007297. - Vladimir Kruchinin, Oct 02 2015
From Peter Bala, Aug 22 2016: (Start)
a(n) = Sum_{k = 0..floor(n/2)} binomial(4*n,n-2*k)*binomial(n+k-1,k).
O.g.f.: A(x) = Hypergeom([5/6, 1/6], [1/2], 108*x^2) + 4*x*Hypergeom([4/3, 2/3], [3/2], 108*x^2).
The o.g.f. is the diagonal of the bivariate rational function 1/(1 - t*(1 + x)^3/(1 - x)) and hence is algebraic by Stanley 1999, Theorem 6.33, p. 197. (End)
a(n) ~ 2^n*3^(3*n/2)/sqrt(2*Pi*n). - Ilya Gutkovskiy, Aug 22 2016
a(n) = 4^n*2*(n+1)*binomial((3*n-1)/2, n+1)/(n-1) for n >= 2. - Peter Luschny, Feb 03 2020
From Peter Bala, Mar 04 2022: (Start)
The o.g.f. A(x) satisfies the algebraic equation (1 - 108*x^2)*A(x)^3 - A(x) = 8*x. Cf. A244039.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k.
Conjecture: the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for primes p >= 5 and positive integers n and k. (End)
From Seiichi Manyama, Aug 09 2025: (Start)
a(n) = [x^n] 1/((1-x)^(n+1) * (1-2*x)^n).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k-1,k) * binomial(2*n-k,n-k).
a(n) = 4^n * binomial((3*n-1)/2,n).
a(n) = [x^n] 1/(1-4*x)^((n+1)/2).
a(n) = [x^n] (1+4*x)^((3*n-1)/2). (End)

A078532 Coefficients of power series that satisfies A(x)^3 - 9xA(x)^4 = 1, A(0)=1.

Original entry on oeis.org

1, 3, 27, 315, 4158, 59049, 880308, 13586859, 215233605, 3479417370, 57168561996, 951892141473, 16026585711660, 272383068872700, 4666865660812044, 80521573261807755, 1397858693681272230, 24398716826612190447, 427921056863230599900, 7537621933880388620010

Offset: 0

Paul D. Hanna, Nov 28 2002

If A(x) = Sum_{k>=1} a(k)x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(n-1) = n^(2n-3) and a(2n-1) = n^(4n-2) (conjecture).
If A(x) = Sum_{k>=1} a(k)x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(k)=n^(2k)*binomial(k/n+1/n+k-1,k)/(k+1) and, consequently, a(n-1) = n^(2n-3) and a(2n-1) = n^(4n-2). - Emeric Deutsch, Dec 10 2002
A generalization of the Catalan sequence (A000108) since for n = 1 the equation A(x)^n -(n^2)*x*A(x)^(n+1) = 1 reduces to A(x)=1+xA(x)^2. - Emeric Deutsch, Dec 10 2002
Radius of convergence of g.f. A(x) is r = 1/(3*4^(4/3)) where A(r) = 4^(1/3). - Paul D. Hanna, Jul 24 2012
Self-convolution cube yields A214668.

			A(x)^3 - 9x*A(x)^4 = 1 since A(x)^3 = 1 +9x +108x^2 +1458x^3 +21060x^4 +... and A(x)^4 = 1 +12x +162x^2 +2340x^3 +... also a(2)=3^3, a(5)=3^10.

G. C. Greubel, Table of n, a(n) for n = 0..750

Cf. A214668, A078531, A078533, A078534, A078535.
Cf. A004987, A008931, A245114, A247029, A376282, A376636.
Cf. A004990.

Table[3^(2n) Binomial[(4n-2)/3,n]/(n+1),{n,0,20}] (* Harvey P. Dale, Nov 03 2011 *)

for(n=0,25, print1(9^n * binomial((4*n-2)/3, n)/(n+1), ", ")) \\ G. C. Greubel, Jan 26 2017

a(n) = 3^(2n)*binomial(4n/3-2/3, n)/(n+1). - Emeric Deutsch, Dec 10 2002
Sequence with offset 1 is expansion of reversion of g.f. x*(1-9*x)^(1/3), which equals x times the g.f. of A004990.
a(n) ~ 2^(8*n/3-5/6) * 3^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Dec 03 2014
D-finite with recurrence n*(n-1)*(n+1)*a(n) -216*(4*n-5)*(2*n-1)*(4*n-11)*a(n-3)=0. - R. J. Mathar, Mar 24 2023
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^5). - Seiichi Manyama, Jun 20 2025

More terms from Harvey P. Dale, Nov 03 2011

A078534 Coefficients of power series that satisfies A(x)^5 - 25xA(x)^6 = 1, A(0)=1.

Original entry on oeis.org

1, 5, 100, 2625, 78125, 2502500, 84150000, 2929265625, 104646953125, 3814697265625, 141323284375000, 5305403695312500, 201382633183593750, 7715985752343750000, 298023223876953125000, 11591412585295166015625, 453601640704152832031250

Offset: 0

Paul D. Hanna, Nov 28 2002

nonn

If A(x) = Sum_{k>=1} a(k)*x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(n-1) = n^(2*n-3) and a(2*n-1) = n^(4*n-2) (conjecture).
From Emeric Deutsch, Dec 10 2002: (Start)
If A(x) = Sum_{k>=1} a(k)*x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(k) = n^(2*k)*binomial(k/n+1/n+k-1,k)/(k+1) and, consequently, a(n-1) = n^(2*n-3) and a(2*n-1) = n^(4*n-2).
A generalization of the Catalan sequence (A000108) since for n = 1 the equation A(x)^n - (n^2)*x*A(x)^(n+1) = 1 reduces to A(x) = 1 + x*A(x)^2. (End)

			A(x)^5 - 25*x*A(x)^6 = 1 since A(x)^5 = 1 + 25*x + 750*x^2 + 24375*x^3 + 831250*x^4 + ... and A(x)^6 = 1 + 30*x + 975*x^2 + 33250*x^3 + ... also a(4) = 5^7, a(9) = 5^18 = 3814697265625.

G. C. Greubel, Table of n, a(n) for n = 0..610

Cf. A078531, A078532, A078533, A078535.

Table[5^(2n) Binomial[(6n-4)/5,n]/(n+1),{n,0,25}]  (* Harvey P. Dale, Mar 27 2011 *)

for(n=0,50, print1(5^(2*n)*binomial((6*n-4)/5, n)/(n+1), ", ")) \\ G. C. Greubel, Jan 30 2017

a(n) = 5^(2*n)*binomial(6*n/5 - 4/5, n)/(n+1). - Emeric Deutsch, Dec 10 2002
a(n) ~ sqrt(3) * 6^(6*n/5 - 4/5) * 5^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Dec 03 2014
From Seiichi Manyama, Jun 21 2025: (Start)
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^7).
G.f.: ( (1/x) * Series_Reversion(x/(1+25*x)^(6/5)) )^(1/6). (End)

More terms from Harvey P. Dale, Mar 27 2011

A182122 Expansion of exp( arcsinh( 2*x ) ).

Original entry on oeis.org

1, 2, 2, 0, -2, 0, 4, 0, -10, 0, 28, 0, -84, 0, 264, 0, -858, 0, 2860, 0, -9724, 0, 33592, 0, -117572, 0, 416024, 0, -1485800, 0, 5348880, 0, -19389690, 0, 70715340, 0, -259289580, 0, 955277400, 0, -3534526380, 0, 13128240840, 0, -48932534040, 0, 182965127280

Offset: 0

Michael Somos, Apr 13 2012

sign

			G.f. = 1 + 2*x + 2*x^2 - 2*x^4 + 4*x^6 - 10*x^8 + 28*x^10 - 84*x^12 + ...

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

Cf. A104624.

Cf. A000984, A048990, A078531, A224884, A245112, A245113.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Exp(Argsinh(2*x)))); // G. C. Greubel, Aug 12 2018

s := proc(n) option remember; `if`(n<2, n+1, -4*(n-2)*s(n-2)/(n+1)) end: A127846 := n -> `if`(n<2,n+1,s(n-1)); seq(A127846(n), n=0..47); # Peter Luschny, Sep 23 2014

CoefficientList[Series[Exp[ArcSinh[2x]],{x,0,50}],x] (* Harvey P. Dale, Aug 18 2012 *)
Table[2 HypergeometricPFQ[{-n+1,2-n},{2},-1],{n,0,46}] (* Peter Luschny, Sep 23 2014 *)

{a(n) = if( n<2, (n>=0) + (n>0), n = n-2; if( n%2, 0, (-1)^(n/2) * 4 * binomial( n, n/2) / (n + 2)))};

{a(n) = if( n<0, 0, polcoeff( sqrt( 1 + 4*x^2 + x*O(x^n) ) + 2*x, n ) )};

{a(n) = my(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = sqrt( 1 + 4*x * A)); polcoeff( A, n))};

def A182122(n):
    if n < 2: return n+1
    if n % 2 == 1: return 0
    return (-1)^(n/2-1)*binomial(n,n/2)/(n-1)
[A182122(n) for n in range(47)] # Peter Luschny, Sep 23 2014

G.f.: 2*x + sqrt( 1 + 4*x^2 ) = 1 / (1 - 2*x / (1 + x / (1 - x / (1 + x / ... )))).
The g.f. A(x) satisfies: A(x) = sqrt(1 + 4*x * A(x)).
a(n) = (-1)^n * A104624(n). Convolution inverse of A104624.
Conjecture : n*(n+1)*a(n) + (n+2)*(n-1)*a(n-1) +4*(n+1)*(n-3)*a(n-2) +4*(n+2)*(n-4)*a(n-3) = 0.- R. J. Mathar, Jul 24 2012
a(n) = 2*hypergeom([-n+1,2-n],[2],-1). - Peter Luschny, Sep 23 2014
0 = a(n)*(+16*a(n+2) + 10*a(n+4)) + a(n+2)*(-2*a(n+2) + a(n+4)) if n>=0. - Michael Somos, Jan 10 2017
a(n+4) = 2 * a(n+2) * (a(n+2) - 8*a(n)) / (a(n+2) + 10*a(n)) if n>=0 is even. - Michael Somos, Jan 10 2017
G.f. A(x) satisfies A(x) = 1/A(-x). - Seiichi Manyama, Jun 20 2025

A214377 G.f. A(x) satisfies A(x) = 1 + 4xA(x)^(3/2).

Original entry on oeis.org

1, 4, 24, 168, 1280, 10296, 86016, 739024, 6488064, 57946200, 524812288, 4808643120, 44493176832, 415146189360, 3901709352960, 36902658748320, 350980432461824, 3354743017001880, 32207616155320320, 310446853795570800, 3003167577200394240, 29146910264615460240

Offset: 0

Paul D. Hanna, Jul 14 2012

nonn

Radius of convergence of g.f. A(x) is r = sqrt(3)/18 where A(r) = 3.

			G.f.: A(x) = 1 + 4*x + 24*x^2 + 168*x^3 + 1280*x^4 + 10296*x^5 + 86016*x^6 + ... where A(x) = 1 + 4*x*A(x)^(3/2).
Radius of convergence: r = 1/(2*3^(3/2)) = 0.09622504486...
Related expansions:
A(x)^(3/2) = 1 + 6*x + 42*x^2 + 320*x^3 + 2574*x^4 + 21504*x^5 + 184756*x^6 + ...
A(x)^(1/2) = 1 + 2*x + 10*x^2 + 64*x^3 + 462*x^4 + 3584*x^5 + 29172*x^6 + ... + A078531(n)*x^n + ...

Bruce C. Berndt, Ramanujan's Notebooks Part I, Springer Verlag, 1985, p. 305.

G. C. Greubel, Table of n, a(n) for n = 0..975
Alois Panholzer, Parking function varieties for combinatorial tree models, arXiv:2007.14676 [math.CO], 2020.

Cf. A078531, A135863, A214553, A098616.

Table[4^n*Binomial[3*n/2, n]*2/(n+2), {n, 0, 20}] (* Vaclav Kotesovec, Oct 20 2015 *)

{a(n)=4^n*binomial(3/2*n,n)/(n/2+1)}
for(n=0,30,print1(a(n),", "))

a(n) = 2^(2*n+1) * binomial(3*n/2, n) / (n+2).
Self-convolution of A078531.
A(-x) = 1/x * series reversion( x*(2*x + sqrt(1 + 4*x^2))^2 ) follows from the Lagrange inversion formula and equation 1.13, p. 305 in Berndt. Cf. A098616. - Peter Bala, Oct 19 2015
a(n) ~ 2^(n + 1/2) * 3^(3*n/2 + 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 20 2015
G.f.: 4*x*(sin(asin(216*x^2-1)/3)/(6*x)+1/(12*x))^3+1. - Vladimir Kruchinin, Sep 30 2022
From Paul D. Hanna, Feb 03 2023: (Start)
G.f. A(x) satisfies: A(x) = 1/(1 - 4*x*A(x)^(1/2)).
G.f. A(x) satisfies: A(x) = B(x*A(x)) where B(x) = A(x/B(x)) = 1 + 8*x^2 + 4*x*sqrt(1 + 4*x^2) is the g.f. of A135863. (End)
From Karol A. Penson, Mar 23 2024: (Start)
G.f. = 1/(48*z^2) - 2F1([-2/3, -1/3], [-1/2], 108*z^2)/(48*z^2) + 4*z*2F1([5/6, 7/6],[5/2],108*z^2);                                     a(n) = Integral_{x=0..sqrt(108)} x^n*W(x), with W(x) = ((72*(g1(x) - g2(x)) + x^2*(-g1(x) + g2(x)) + 4*sqrt(-3*x^2 + 324)*(g1(x) + g2(x)))*3^(1/6))/(96*Pi*(x^2)^(5/6)),
  where g1(x) = (18 - sqrt(324 - 3*x^2))^(2/3) and
  g2(x) = (18 + sqrt(324 - 3*x^2))^(2/3).
This integral representation is unique as W(x) is the solution of the Hausdorff power moment problem on x = (0, sqrt(108)). Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0, with singularity x^(-1/3), and for x > 0 is monotonically decreasing to zero at x = sqrt(108). For x -> sqrt(108), W'(x) tends to -infinity. (End)
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^2). - Seiichi Manyama, Jun 17 2025
D-finite with recurrence -(n+2)*(n-1)*a(n) +12*(3*n-2)*(3*n-4)*a(n-2)=0. - R. J. Mathar, Jul 30 2025

A078533 Coefficients of power series that satisfies A(x)^4 - 16x*A(x)^5 = 1, A(0)=1.

Original entry on oeis.org

1, 4, 56, 1024, 21216, 473088, 11075328, 268435456, 6677665280, 169514369024, 4373549027328, 114349209288704, 3023068543631360, 80675644291153920, 2170389180446539776, 58798996734949195776, 1602737048880933109760, 43924199383151211970560

Offset: 0

Paul D. Hanna, Nov 28 2002

nonn

If A(x) = Sum_{k>=1} a(k)x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(n-1) = n^(2n-3) and a(2n-1) = n^(4n-2) (conjecture).
If A(x) = Sum_{k>=1} a(k)x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(k) = n^(2k)*binomial(k/n + 1/n + k - 1, k)/(k+1) and, consequently, a(n-1) = n^(2n-3) and a(2n-1) = n^(4n-2). - Emeric Deutsch, Dec 10 2002
A generalization of the Catalan sequence (A000108) since for n = 1 the equation A(x)^n - (n^2)*x*A(x)^(n+1) = 1 reduces to A(x) = 1 + xA(x)^2. - Emeric Deutsch, Dec 10 2002

			A(x)^4 - 16x*A(x)^5 = 1 since A(x)^4 = 1 + 16x + 320x^2 + 7040x^3 + 163840x^4 + ... and A(x)^5 = 1 + 20x + 440x^2 + 10240x^3 + ... also a(3) = 4^5, a(7) = 4^14 = 268435456.

G. C. Greubel, Table of n, a(n) for n = 0..675

Cf. A078531, A078532, A078534, A078535.

Table[4^(2*n)*Binomial[5*n/4-3/4, n]/(n+1),{n,0,20}] (* Vaclav Kotesovec, Dec 03 2014 *)

for(n=0,50, print1(2^(4*n)*binomial((5*n-3)/4,n)/(n+1), ", ")) \\ G. C. Greubel, Jan 30 2017

a(n) = 4^(2n)*binomial(5n/4 - 3/4, n)/(n+1). - Emeric Deutsch, Dec 10 2002
a(n) ~ 5^(5*n/4 - 1/4) * 2^(2*n - 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Dec 03 2014
From Seiichi Manyama, Jun 21 2025: (Start)
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^6).
G.f.: ( (1/x) * Series_Reversion(x/(1+16*x)^(5/4)) )^(1/5). (End)

A078535 Coefficients of power series that satisfies A(x)^6 - 36x*A(x)^7 = 1, A(0)=1.

Original entry on oeis.org

1, 6, 162, 5760, 232254, 10077696, 458960580, 21634449408, 1046465787510, 51644846702592, 2590092194793948, 131621703842267136, 6762649550214036780, 350714987252652441600, 18334388441036020419720, 965148007553698721955840, 51116742846877582931249574

Offset: 0

Paul D. Hanna, Nov 28 2002

nonn

If A(x)=sum_{k=1..inf} a(k)x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(n-1) = n^(2n-3) and a(2n-1) = n^(4n-2) (conjecture).
If A(x)=sum_{k=1..inf} a(k)x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(k)=n^(2k)*binomial(k/n+1/n+k-1,k)/(k+1) and, consequently, a(n-1) = n^(2n-3) and a(2n-1) = n^(4n-2). - Emeric Deutsch, Dec 10 2002
A generalization of the Catalan sequence (A000108) since for n = 1 the equation A(x)^n -(n^2)*x*A(x)^(n+1) = 1 reduces to A(x)=1+xA(x)^2. - Emeric Deutsch, Dec 10 2002

			A(x)^6 - 36x*A(x)^7 = 1 since A(x)^6 = 1 +36x +1512x^2 +68040x^3 +3193344x^4 +... and A(x)^7 = 1 +42x +1890x^2 +88704x^3 +... also a(5)=6^9, a(11)=6^22 = 131621703842267136.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A078531, A078532, A078533, A078534.

Table[6^(2*n)*Binomial[7*n/6-5/6, n]/(n+1),{n,0,20}] (* Vaclav Kotesovec, Dec 03 2014 *)

a(n) = {6^(2*n)*binomial((7*n-5)/6, n)/(n+1)} \\ Andrew Howroyd, Nov 05 2019

a(n) = 6^(2n)*binomial(7n/6-5/6, n)/(n+1). - Emeric Deutsch, Dec 10 2002
a(n) ~ 7^(7*n/6-1/3) * 6^n / (sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Dec 03 2014
From Seiichi Manyama, Jun 21 2025: (Start)
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^8).
G.f.: ( (1/x) * Series_Reversion(x/(1+36*x)^(7/6)) )^(1/7). (End)

Terms a(13) and beyond from Andrew Howroyd, Nov 05 2019

A138020 G.f. satisfies A(x) = sqrt( (1 + 2xA(x)) / (1 - 2xA(x)) ).

Original entry on oeis.org

1, 2, 6, 24, 110, 544, 2828, 15232, 84246, 475648, 2730068, 15882240, 93438540, 554967040, 3323125528, 20039827456, 121597985254, 741871845376, 4548193111428, 28004975116288, 173113004348580, 1073893324357632, 6683288759506856, 41715337804120064

Offset: 0

Paul D. Hanna, Feb 28 2008

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A078531, A151374.

A138020 := proc(n)
    option remember ;
    if n < 5 then
        op(n+1,[1,2,6,24,110]) ;
    else
        4*(-55*n^3 +231*n^2 -263*n +51)*procname(n-2) -16*(n-3)*(n-4)*(5*n-1)*procname(n-4) ;
        -%/n/(n+1)/(5*n-11)
    end if;
end proc:
seq(A138020(n),n=0..30) ; # R. J. Mathar, Sep 27 2024

CoefficientList[y/.AsymptoticSolve[y^2-1-2x(y+y^3) ==0,y->1,{x,0,23}][[1]],x] (* Alexander Burstein, Nov 26 2021 *)

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

a(n) ~ 2^(n - 1/2) * phi^((5*n + 3)/2) / (sqrt(Pi) * 5^(1/4) * n^(3/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 04 2020
From Alexander Burstein, Nov 26 2021: (Start)
G.f.: A(x) = 1 + 2*x*A(x)*(1 + A(x)^2)/(1 + A(x)).
G.f.: A(-x*A(x)^2) = 1/A(x). (End)
D-finite with recurrence +n*(n+1)*(5*n-11) *a(n) +4*(-55*n^3 +231*n^2 -263*n +51)*a(n-2) -16*(n-3)*(n-4)*(5*n-1)*a(n-4)=0. - R. J. Mathar, Mar 25 2024
From Seiichi Manyama, Dec 22 2024: (Start)
a(n) = (2^n/(n+1)) * Sum_{k=0..n} binomial(n/2+k-1/2,k) * binomial(n/2+1/2,n-k).
a(n) = 2^n * Sum_{k=0..n} binomial(n,k) * binomial(n/2+k+1/2,n)/(n+2*k+1). (End)

cp's OEIS Frontend

A007297 Number of connected graphs on n labeled nodes on a circle with straight-line edges that don't cross.

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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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A091527 a(n) = ((3*n)!/n!^2)*(Gamma(1+n/2)/Gamma(1+3n/2)).

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A078532 Coefficients of power series that satisfies A(x)^3 - 9*x*A(x)^4 = 1, A(0)=1.

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A078534 Coefficients of power series that satisfies A(x)^5 - 25*x*A(x)^6 = 1, A(0)=1.

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A182122 Expansion of exp( arcsinh( 2*x ) ).

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

A091527 a(n) = ((3n)!/n!^2)(Gamma(1+n/2)/Gamma(1+3n/2)).

A078532 Coefficients of power series that satisfies A(x)^3 - 9xA(x)^4 = 1, A(0)=1.

A078534 Coefficients of power series that satisfies A(x)^5 - 25xA(x)^6 = 1, A(0)=1.

A214377 G.f. A(x) satisfies A(x) = 1 + 4xA(x)^(3/2).

A138020 G.f. satisfies A(x) = sqrt( (1 + 2xA(x)) / (1 - 2xA(x)) ).