A224884 -id:A224884 - cp's OEIS frontend

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

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

A085614 Number of elementary arches of size n.

Original entry on oeis.org

1, 3, 16, 105, 768, 6006, 49152, 415701, 3604480, 31870410, 286261248, 2604681690, 23957864448, 222399744300, 2080911654912, 19604537460045, 185813170126848, 1770558814528770, 16951376923852800, 162984598242674670

Offset: 1

N. J. A. Sloane, Jul 10 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..200
F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).
Karl Dilcher, Armin Straub, andChristophe Vignat, Identities for Bernoulli polynomials related to multiple Tornheim zeta functions, arXiv:1903.11759 [math.NT], 2019. See p. 13.
Loïc Foissy, Free quadri-algebras and dual quadri-algebras, arXiv preprint 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.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Vincent Pilaud, Pebble trees, arXiv:2205.06686 [math.CO], 2022.
Thomas M. Richardson, The three 'R's and Dual Riordan Arrays, arXiv:1609.01193 [math.CO], 2016.
M. R. Sepanski, On Divisibility of Convolutions of Central Binomial Coefficients, Electronic Journal of Combinatorics, 21 (1) 2014, #P1.32.
Jian Zhou, Fat and Thin Emergent Geometries of Hermitian One-Matrix Models, arXiv:1810.03883 [math-ph], 2018.

Cf. A143018, A206300.

with(combstruct); ar := {EA = Union(Sequence(EA, card >= 2), Prod(Z, Sequence(EA), Sequence(EA))), C=Union(Z, Prod(Z,Z,Sequence(EA), Sequence(EA), Sequence(Union(Sequence(EA,card>=1), Prod(Z,Sequence(EA),Sequence(EA))))))}; seq(count([EA,ar], size=i),i=1..20);

Rest[CoefficientList[Series[1/6*Sqrt[3]*Sin[1/3*ArcSin[6*Sqrt[3]*x]] - 1/2*Cos[1/3*ArcSin[6*Sqrt[3]*x]],{x,0,20}],x]] (* Vaclav Kotesovec, Oct 21 2012 *)
Rest[CoefficientList[InverseSeries[Series[x - 3*x^2 + 2*x^3, {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Aug 22 2017 *)
(* From Dixon J. Jones, Apr 15 2021: (Start) *)
Table[4^n Gamma[(3n + 2)/2]/(Gamma[(n + 2)/2](n + 1)!), {n, 0, 20}]
Table[4^n Pochhammer[(n + 2)/2, n]/(n + 1)!, {n, 0, 20}] (* End *)

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

G.f. is the series reversion of x-3*x^2+2*x^3.
a(n) = 2^n*(3*n)!!/((n+1)!*n!!). - Maxim Krikun (krikun(AT)iecn.u-nancy.fr), May 25 2007
G.f.: 1/6*sqrt(3)*sin(1/3*arcsin(6*sqrt(3)*x))-1/2*cos(1/3*arcsin(6*sqrt(3)*x)). - Vaclav Kotesovec, Oct 21 2012
Conjecture: n*(n-1)*a(n) +(n-1)*(n-2)*a(n-1) -12*(3*n-5)*(3*n-7)*a(n-2) -12*(3*n-8)*(3*n-10)*a(n-3) = 0. - R. J. Mathar, Oct 18 2013
a(n) ~ 2^(n - 3/2) * 3^(3*n/2 - 1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2017
From Dixon J. Jones, Apr 15 2021: (Start)
a(n) = A206300(n)/2 = abs(A224884(n))/2 for n>=1.
a(n) = 4^n Gamma((3*n + 2)/2)/(Gamma((n + 2)/2)*(n + 1)!).
a(n) = (4^n*((n + 2)/2)_n)/(n + 1)!, where (x)_k is the Pochhammer symbol. (End)

A247029 G.f. A(x) satisfies A(x) = A(x)^4 - 9*x.

Original entry on oeis.org

1, 3, -18, 180, -2187, 29484, -424116, 6377292, -99034650, 1576075644, -25569752274, 421325812440, -7031733125508, 118620405322020, -2019349799669160, 34647126360607440, -598525520999144643, 10401492640172342940, -181721630178565389900, 3189811189331825319492

Offset: 0

Paul D. Hanna, Sep 09 2014

sign

			G.f.: A(x) = 1 + 3*x - 18*x^2 + 180*x^3 - 2187*x^4 + 29484*x^5 - 424116*x^6 +...
where
A(x)^4 = 1 + 12*x - 18*x^2 + 180*x^3 - 2187*x^4 + 29484*x^5 - 424116*x^6 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A004987, A008931, A078532, A245114, A376282, A376636.

Cf. A224884.

FullSimplify[Table[-(-1)^n * 3^(2*n-1) * 4^(n-1) * Gamma[n/3 + 1/6] * Gamma[2*n/3 - 1/6] / (Pi * Gamma[n + 1]), {n, 0, 20}]] (* Vaclav Kotesovec, Nov 18 2017 *)

{a(n)=polcoeff(x/serreverse(x*(1+9*x +x^2*O(x^n))^(1/3)), n)}
for(n=0, 25, print1(a(n), ", "))

G.f.: x / Series_Reversion( x*(1 + 9*x)^(1/3) ).
Recurrence: (n-2)*(n-1)*n*a(n) = -216*(2*n - 5)*(4*n - 13)*(4*n - 7)*a(n-3). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ -(-1)^n * 2^(8*n/3 - 13/6) * 3^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Nov 18 2017
G.f. A(x) satisfies A(x) = 1/A(-x/A(x)^5). - Seiichi Manyama, Jun 20 2025

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

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

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A085614 Number of elementary arches of size n.

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A247029 G.f. A(x) satisfies A(x) = A(x)^4 - 9*x.

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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.

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