A195699 -id:A195699 - cp's OEIS frontend

A003645 a(n) = 2^n * C(n+1), where C(n) = A000108(n) Catalan numbers.

1, 4, 20, 112, 672, 4224, 27456, 183040, 1244672, 8599552, 60196864, 426008576, 3042918400, 21909012480, 158840340480, 1158600130560, 8496400957440, 62605059686400, 463277441679360, 3441489566760960, 25654740406763520, 191852841302753280, 1438896309770649600

Offset: 0

N. J. A. Sloane

Number of nonisomorphic unrooted unicursal planar maps with n+2 edges and exactly one vertex of valency 1 (unicursal means that exactly two vertices are of odd valency). - Valery A. Liskovets, Apr 07 2002
Total number of vertices in rooted Eulerian planar maps with n+1 edges.
Half the number of ways to dog-ear every page of an (n+1)-page book. - R. H. Hardin, Jun 21 2002
Convolution of A052701(n+1) with itself.
Number of Motzkin lattice paths with weights: 1 for up step, 4 for level step and 4 for down step. - Wenjin Woan, Oct 24 2004
The number of rooted bipartite n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005
Also the number of paths of length 2n+1 in a binary tree between two vertices that are one step away from each other. - David Koslicki (koslicki(AT)math.psu.edu), Nov 02 2010
2*a(n) for n > 1 is the number of increasing strict binary trees with 2n-1 nodes that simultaneously avoid 213 and 231 in the classical sense. For more information about increasing strict binary trees with an associated permutation, see A245894. - Manda Riehl, Aug 22 2014

L. M. Koganov, V. A. Liskovets, T. R. S. Walsh, Total vertex enumeration in rooted planar maps, Ars Combin. 54 (2000), 149-160.
V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

A. Claesson, S. Kitaev and A. de Mier, An involution on bicubic maps and beta(0,1)-trees, arXiv preprint arXiv:1210.3219 [math.CO], 2012. - From _N. J. A. Sloane_, Jan 01 2013
S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, 2017.
Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 652.
Anatol N. Kirillov, Notes on Schubert, Grothendieck and key polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 034, 56 p. (2016).
Huyile Liang, Jeffrey Remmel, and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see page 13.
V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., Vol. 282, No. 1-3 (2004), pp. 209-221.
V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., Vol. 36, No.4 (2006), pp. 364-387.
Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 11.
Youngja Park and SeungKyung Park, Enumeration of generalized lattice paths by string types, peaks, and ascents, Discrete Mathematics, Vol. 339, No. 11 (2016), pp. 2652-2659.
M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seq., Vol. 9 (2006), Article 06.1.1.

Cf. A069724, A069725, A090375.
Third row of array A102539.
Column of array A073165.
Cf. A000108, A052701, A195699.

[2^n*Binomial(2*n+3, n+1)/(2*n+3) : n in [0..30]]; // Wesley Ivan Hurt, Aug 23 2014

A003645:=n->2^n*binomial(2*n+3, n+1)/(2*n+3): seq(A003645(n), n=0..30); # Wesley Ivan Hurt, Aug 23 2014

Table[2^n CatalanNumber[n+1],{n,0,20}] (* Harvey P. Dale, May 07 2013 *)

a(n)=if(n<0,0,2^n*(2*n+2)!/(n+1)!/(n+2)!)

a(n) = A052701(n+2)/2.
2*a(n) matches the odd-indexed terms of A090375.
a(n) = 2^n * binomial(2n+3, n+1) / (2n+3). - Len Smiley, Feb 24 2006
G.f.: (1-4x-sqrt(1-8x))/(8x^2) = C(2x)^2, where C(x) is the g.f. for Catalan numbers, A000108.
From Gary W. Adamson, Jul 12 2011: (Start)
Let M = the following production matrix:
  2, 2, 0, 0, 0, ...
  2, 2, 2, 0, 0, ...
  2, 2, 2, 2, 0, ...
  2, 2, 2, 2, 2, ...
  ...
a(n) = sum of top row terms in M^n. Example: top row of M^3 = (40, 40, 24, 8, 0, 0, 0, ...), sum = 112 = a(3). (End)
D-finite with recurrence (n+2)*a(n) - 4*(2n+1)*a(n-1) = 0. - R. J. Mathar, Apr 01 2012
E.g.f.: a(n) = n!* [x^n] exp(4*x)*BesselI(1, 4*x)/(2*x). - Peter Luschny, Aug 25 2012
Expansion of square of continued fraction 1/(1 - 2*x/(1 - 2*x/(1 - 2*x/(1 - ...)))). - Ilya Gutkovskiy, Apr 19 2017
From Amiram Eldar, Mar 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 38/49 + 192*arcsin(sqrt(1/8))/(49*sqrt(7)).
Sum_{n>=0} (-1)^n/a(n) = 14/27 + 32*log(2)/81. (End)
a(n) = Product_{1 <= i <= j <= n} (i + j + 2)/(i + j - 1). Cf. A001700. - Peter Bala, Feb 22 2023

A052701 a(0) = 0; for n >= 1, a(n) = 2^(n-1)*C(n-1), where C(n) = A000108(n) Catalan numbers, n>0.

Original entry on oeis.org

0, 1, 2, 8, 40, 224, 1344, 8448, 54912, 366080, 2489344, 17199104, 120393728, 852017152, 6085836800, 43818024960, 317680680960, 2317200261120, 16992801914880, 125210119372800, 926554883358720, 6882979133521920

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

A151374 shifted one place right. - Joerg Arndt, Mar 17 2011
The number of rooted Eulerian n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005
This is also the number of strings of length 2n-2 of two different types of balanced parentheses. For example, a(2) = 2, since the two possible strings of length 2 are [] and (), a(3) = 8, since the 8 possible strings of length 4 are (()), [()], ([]), [[]], ()(), [](), ()[], and [][]. - Jeffrey Shallit, Jun 03 2006
Row sums of number triangle A110506. - Paul Barry, Jul 24 2005
Also row sums of triangle in A085880. - Philippe Deléham, Aug 01 2005
Row sums of number triangle A114608. - Philippe Deléham, Oct 15 2008

V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Sergey Kirgizov, and Mehdi Naima, A lattice on Dyck paths close to the Tamari lattice, arXiv:2309.00426 [math.CO], 2023.
M. Bousquet-Mélou, Limit laws for embedded trees, arXiv:math/0501266 [math.CO], 2005.
Marek Bożejko, Maciej Dołęga, Wiktor Ejsmont, and Światosław R. Gal, Reflection length with two parameters in the asymptotic representation theory of type B/C and applications, arXiv:2104.14530 [math.RT], 2021.
F. Chapoton and S. Giraudo, Enveloping operads and bicoloured noncrossing configurations, arXiv:1310.4521 [math.CO], 2013.
Ivan Geffner and Marc Noy, Counting Outerplanar Maps, Electronic Journal of Combinatorics, Vol. 24, No. 2 (2017), Article P2.3.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 651.
V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., Vol. 36, No.4 (2006), pp. 364-387.
Vincent Pilaud and V. Pons, Permutrees, arXiv preprint arXiv:1606.09643 [math.CO], 2016-2017.
Index to sequences related to reversion of series.

Limit of array A102544.

Cf. A000108, A003645, A025225, A052714, A195699.

spec := [S,{B=Union(C,Z),S=Union(B,C),C=Prod(S,S)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

InverseSeries[Series[y-2*y^2, {y, 0, 24}], x] (* then A(x)=y(x) *) (* Len Smiley, Apr 07 2000 *)
Join[{0},Table[2^n CatalanNumber[n],{n,0,30}]] (* Harvey P. Dale, Aug 29 2015 *)

a(n)=if(n<1,0,2^(n-1)*(2*n-2)!/(n-1)!/n!)

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

a(n)=if(n<1,0,polcoeff(2*x/(1+sqrt(1-8*x+O(x^n))),n))

a(n) = A052714(n)/n!.
a(n) = A003645(n-2)*2, n>1.
a(n) = 8^(n-1)*GAMMA(n-1/2)/GAMMA(n+1)/Pi^(1/2), n>0.
D-finite with recurrence: a(1)=1, (-4+8*n)*a(n) - (n+1)*a(n+1) = 0.
G.f.: (1-sqrt(1-8*x))/4 = x*C(2*x) where C(x) is g.f. for Catalan numbers, A000108.
G.f. A(x) satisfies 2*A(x)^2-A(x)+x=0, A(0)=0 and A(x)=x+2*A(x)^2=x/(1-2*A(x)). Series reversion of x-2*x^2. - Michael Somos, Sep 06 2003
a(0)=0, a(1)=1; a(n) = 2*Sum_{i=1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
With a different offset, a(0)=1, a(n) = Sum_{k=0..n} Sum_{j=0..n} (j*C(2n-j-1, n-j)*C(j, k)*2^(n-j)/n), n>0. - Paul Barry, Jul 24 2005
The Hankel transform of a(n+1) = [1,2,8,40,224,1344,...] is 4^C(n+1,2). - Philippe Deléham, Nov 06 2007
G.f.: x + 4*x^2/(G(0)-4*x) where G(k) = k*(8*x+1) + 4*x + 2 - 2*x*(2*k+3)*(2*k+4)/G(k+1) ; (continued fraction ). - Sergei N. Gladkovskii, Apr 05 2013
a(n) ~ 8^(n-1)/(sqrt(Pi)*n^(3/2)). - Ilya Gutkovskiy, Dec 04 2016
From Amiram Eldar, Mar 06 2022: (Start)
Sum_{n>=1} 1/a(n) = 68/49 + 96*arcsin(sqrt(1/8))/(49*sqrt(7)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 20/27 - 16*log(2)/81. (End)
a(n) = A025225(n)/2 for n>=1. - Alois P. Heinz, Feb 16 2025

Better description from Claude Lenormand (claude.lenormand(AT)free.fr), Mar 19 2001

Additional comments from Michael Somos, Feb 24 2002

A034910 One quarter of octo-factorial numbers.

Original entry on oeis.org

1, 12, 240, 6720, 241920, 10644480, 553512960, 33210777600, 2258332876800, 171633298636800, 14417197085491200, 1326382131865190400, 132638213186519040000, 14324927024144056320000, 1661691534800710533120000, 206049750315288106106880000

Offset: 1

Wolfdieter Lang

A034910 occurs in connection with the Vandermonde permanent of (1,3,5,7,9,...); see the Mathematica section of A203516. - Clark Kimberling, Jan 03 2012

			G.f. = x + 12*x^2 + 240*x^3 + 6720*x^4 + 241920*x^5 + 10644480*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 1..325
Index entries for sequences related to factorial numbers.

Cf. A001147, A045755, A034908, A034909, A034911, A034912, A203516.

[n le 2 select 12^(n-1) else (7*n-3)*Self(n-1) +4*(n-1)*(2*n-3)*Self(n-2): n in [1..30]]; // G. C. Greubel, Oct 20 2022

[seq((2*n)!/(n)!*2^(n-2), n=1..14)]; # Zerinvary Lajos, Sep 25 2006

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 11, 5!, 8}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
a[ n_] := Pochhammer[ 1/2, n] 8^n / 4; (* Michael Somos, Feb 04 2015 *)

{a(n) = if( n==1, 1, n>1, a(n-1) * (8*n - 4), a(n+1) / (8*n + 4))}; /* Michael Somos, Feb 04 2015 */

[2^(3*n-2)*rising_factorial(1/2, n) for n in range(1,40)] # G. C. Greubel, Oct 20 2022

4*a(n) = (8*n-4)(!^8) = Product_{j=1..n} (8*j-4) = 4^n*A001147(n) = 2^n*(2*n)!/n!, A001147(n) = (2*n-1)!!.
E.g.f. (-1+(1-8*x)^(-1/2))/4.
a(n) = A090802(2n-1, n). - Ross La Haye, Oct 18 2005
a(n) = ((2*n)!/n!)*2^(n-2). - Zerinvary Lajos, Sep 25 2006
G.f.: x/(1-12*x/(1-8*x/(1-20*x/(1-16*x/(1-28*x/(1-24*x/(1-36*x/(1-32*x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2011
From Peter Bala, Feb 01 2015: (Start)
Recurrence equation: a(n) = (7*n - 3)*a(n-1) + 4*(n - 1)*(2*n - 3)*a(n-2).
The sequence b(n) := a(n)* Sum_{k = 0..n-1} (-1)^k/( 2^k*(2*k + 1)*binomial(2*k,k) ) beginning [1, 11, 222, 6210, 223584, ...] satisfies the same recurrence. This leads to the finite continued fraction expansion b(n)/a(n) = 1/(1 + 1/(11 + 24/(18 + 60/(25 + ... + 4*(n - 1)*(2*n - 3)/(7*n - 3) )))) for n >= 3.
Letting n tend to infinity gives the continued fraction expansion Sum_{k>=0} (-1)^k/( 2^k*(2*k + 1)*binomial(2*k,k) ) = (4/3)*log(2) = 1/(1 + 1/(11 + 24/(18 + 60/(25 + ... + 4*(n - 1)*(2*n - 3)/((7*n - 3) + ... ))))). (End)
From Peter Bala, Feb 03 2015: (Start)
This sequence satisfies several other second order recurrence equations leading to some continued fraction expansions.
1) a(n) = (9*n + 4)*a(n-1) - 4*n*(2*n - 1)*a(n-2).
This recurrence is also satisfied by the (integer) sequence c(n) := a(n)*Sum_{k = 0..n} 1/( 2^k*(2*k + 1)*binomial(2*k,k) ). From this we can obtain the continued fraction expansion Sum_{k >= 0} 1/( 2^k*(2*k + 1)*binomial(2*k,k) ) = (8/sqrt(7))*arctan(sqrt(7)/7) = (8/sqrt(7))*A195699 = 1 + 1/(12 - 24/(22 - 60/(31 - ... - 4*n*(2*n - 1)/((9*n + 4) - ... )))).
2) a(n) = (12*n + 2)*a(n-1) - 8*(2*n - 1)^2*a(n-2).
This recurrence is also satisfied by the (integer) sequence d(n) := a(n)*Sum_{k = 0..n} 1/( (2*k + 1)*2^k ). From this we can obtain the continued fraction expansion Sum_{k >= 0} 1/( (2*k + 1)*2^k ) = (1/sqrt(2))*log(3 + 2*sqrt(2)) = 1 + 2/(12 - 8*3^2/(26 - 8*5^2/(38 - ... - 8*(2*n - 1)^2/((12*n + 2) - ... )))). Cf. A002391.
3) a(n) = (4*n + 6)*a(n-1) + 8*(2*n - 1)^2*a(n-2).
This recurrence is also satisfied by the (integer) sequence e(n) := a(n)*Sum_{k = 0..n} (-1)^k/( (2*k + 1)*2^k ). From this we can obtain the continued fraction expansion Sum_{k >= 0} (-1)^k/( (2*k + 1)*2^k ) = (1/sqrt(2))*arctan(sqrt(2)/2) = 1 - 2/(12 + 8*3^2/(14 + 8*5^2/(18 + ... + 8*(2*n - 1)^2/((4*n + 6) + ... )))). Cf. A073000. (End)
a(n) = (-1)^n / (16*a(-n)) for all n in Z. - Michael Somos, Feb 04 2015
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = e^(1/8)*sqrt(2*Pi)*erf(1/(2*sqrt(2))), where erf is the error function.
Sum_{n>=1} (-1)^(n+1)/a(n) = e^(-1/8)*sqrt(2*Pi)*erfi(1/(2*sqrt(2))), where erfi is the imaginary error function. (End)

A168229 Decimal expansion of arctan(sqrt(7)).

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Nov 20 2009

This constant is the least x > 0 satisfying cos(4*x) = (cos x)^2. - Clark Kimberling, Oct 15 2011

An identity resembling Machin's Pi/4 = arctan(1/1) = 4*arctan(1/5) - arctan(1/239) is arctan(sqrt(7)/1) = 5*arctan(sqrt(7)/11) + 2*arctan(sqrt(7)/181), which can also be expressed as arcsin(sqrt(7/2^3)) = 5*arcsin(sqrt(7/2^7)) + 2*arcsin(sqrt(7/2^15)) (cf. A038198). - Joerg Arndt, Nov 09 2012

			arctan(sqrt(7)) = 1.209429202888189... .

G. C. Greubel, Table of n, a(n) for n = 1..5000
Kunle Adegoke, Infinite arctangent sums involving Fibonacci and Lucas numbers, arXiv:1603.08097 [math.NT], 2016.
Djurdje Cvijovic, A dilogarithmic integral arising in quantum field theory, arXiv:0911.3773 [math.CA], 2009.
Index entries for transcendental numbers

Cf. A000032, A000045, A033891, A038198, A195699.

[Arctan(Sqrt(7))]; // G. C. Greubel, Nov 18 2017

RealDigits[ArcTan[Sqrt[7]], 10, 50][[1]] (* G. C. Greubel, Nov 18 2017 *)

atan(sqrt(7)) \\ Michel Marcus, Mar 11 2013

Smallest positive solution of cos(x) + sqrt(1 + cos^2(x)) = sqrt(2). - Geoffrey Caveney, Apr 24 2014
Equals Sum_{k >= 1} atan(5*sqrt(7)*F(4k-1)/L(2*(4k-1))) where L=A000032 and F=A000045. See also A033891. - Michel Marcus, Mar 29 2016
Equals arccos(1/(2*sqrt(2))). - Amiram Eldar, May 28 2021

More digits from R. J. Mathar, Dec 06 2009

A195704 Decimal expansion of arccos(-sqrt(1/8)).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 23 2011

			arccos(-sqrt(1/8)) = 1.93216345070...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Cf. A195699.

[Arccos(-Sqrt(1/8))]; // G. C. Greubel, Nov 18 2017

r = Sqrt[1/8];
N[ArcSin[r], 100]
RealDigits[%]  (* A195699 *)
N[ArcCos[r], 100]
RealDigits[%]  (* A168229 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A188615 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A195704 *)

acos(-sqrt(1/8)) \\ G. C. Greubel, Nov 18 2017

Equals Pi - arcsin(sqrt(7/2)/2) = Pi - arctan(sqrt(7)). - Amiram Eldar, Jul 09 2023

cp's OEIS Frontend

A003645 a(n) = 2^n * C(n+1), where C(n) = A000108(n) Catalan numbers.

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A052701 a(0) = 0; for n >= 1, a(n) = 2^(n-1)*C(n-1), where C(n) = A000108(n) Catalan numbers, n>0.

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A034910 One quarter of octo-factorial numbers.

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A168229 Decimal expansion of arctan(sqrt(7)).

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A195704 Decimal expansion of arccos(-sqrt(1/8)).

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