A069720 -id:A069720 - cp's OEIS frontend

A069723 a(n) = 2^(n-1)binomial(2n-3, n-1).

1, 2, 12, 80, 560, 4032, 29568, 219648, 1647360, 12446720, 94595072, 722362368, 5538111488, 42600857600, 328635187200, 2541445447680, 19696202219520, 152935217233920, 1189496134041600, 9265548833587200, 72271280901980160, 564404288948797440, 4412615349963325440

Offset: 1

Valery A. Liskovets, Apr 07 2002

Number of rooted unicursal planar maps with n edges and two vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Harlan J. Brothers, Pascal's Prism: Supplementary Material.
Valery A. Liskovets and Timothy R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Main diagonal of array A082137.

Cf. A069720, A069721, A069722, A082143, A082144, A082145, A088218.

Z:=(1-sqrt(1-z))*8^n/sqrt(1-z)/2: Zser:=series(Z, z=0, 33): seq(coeff(Zser, z, n), n=0..20); # Zerinvary Lajos, Jan 16 2007

Table[2^(n - 1) * Binomial[2*n - 3, n - 1], {n, 1,50}] (* G. C. Greubel, Jan 15 2017 *)

# Assuming offset 0:
A069723  = lambda n: (rising_factorial(n, n)/factorial(n)) << n
[A069723(n) for n in (0..20)] # Peter Luschny, Nov 30 2014

a(n) = A069722(n)/2, n>1.
G.f.: 4*x/(sqrt(1-8*x) * (1-sqrt(1-8*x))). - Paul Barry, Sep 06 2004
With offset 0: a(n) = (0^n + 2^n*binomial(2n, n))/2. - Paul Barry, Sep 24 2004
D-finite with recurrence (-n+1)*a(n) + 4*(2*n-3)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012
With offset 0: a(n) = 2^n*rf(n,n)/n! = 2^n*A088218(n), where rf denotes the rising factorial. - Peter Luschny, Nov 30 2014
a(n) = Sum_{k=0..n} binomial(n+k-1,k)*binomial(2*n-1, n-k). - Vladimir Kruchinin, Nov 11 2016
a(n) ~ 2^(3*n-4)/sqrt(Pi*n). - Ilya Gutkovskiy, Nov 11 2016
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=1} 1/a(n) = 9/7 + 16*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 7/9 - 8*log(2)/27. (End)

A082137 Square array of transforms of binomial coefficients, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 4, 1, 4, 12, 16, 8, 1, 5, 20, 40, 40, 16, 1, 6, 30, 80, 120, 96, 32, 1, 7, 42, 140, 280, 336, 224, 64, 1, 8, 56, 224, 560, 896, 896, 512, 128, 1, 9, 72, 336, 1008, 2016, 2688, 2304, 1152, 256, 1, 10, 90, 480, 1680, 4032, 6720, 7680, 5760, 2560, 512

Offset: 0

Paul Barry, Apr 06 2003

Rows are associated with the expansions of (x^k/k!)exp(x)cosh(x) (leading zeros dropped). Rows include A011782, A057711, A080929, A082138, A080951, A082139, A082140, A082141. Columns are of the form 2^(k-1)C(n+k, k). Diagonals include A069723, A082143, A082144, A082145, A069720.
T(n, k) is also the number of idempotent order-preserving and order-decreasing partial transformations (of an n-chain) of width k (width(alpha)= |Dom(alpha)|). - Abdullahi Umar, Oct 02 2008
Read as a triangle this is A119468 with rows reversed. A119468 has e.g.f. exp(z*x)/(1-tanh(x)). - Peter Luschny, Aug 01 2012
Read as a triangle this is a subtriangle of A198793. - Philippe Deléham, Nov 10 2013

			Rows begin
  1 1  2   4   8 ...
  1 2  6  16  40 ...
  1 3 12  40 120 ...
  1 4 20  80 280 ...
  1 5 30 140 560 ...
Read as a triangle, this begins:
  1
  1, 1
  1, 2,  2
  1, 3,  6,  4
  1, 4, 12, 16,   8
  1, 5, 20, 40,  40, 16
  1, 6, 30, 80, 120, 96, 32
  ... - _Philippe Deléham_, Nov 10 2013

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
Laradji, A. and Umar, A. Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8.

Cf. A119468, A007318, A134309.

# As a triangular array:
T := (n,k) -> 2^(k+0^k-1)*binomial(n,k):
for n from 0 to 9 do seq(T(n,k), k=0..n) od; # Peter Luschny, Nov 10 2017

rows = 11; t[n_, k_] := 2^(n-1)*(n+k)!/(n!*k!); t[0, ] = 1; tkn = Table[ t[n, k], {k, 0, rows}, {n, 0, rows}]; Flatten[ Table[ tkn[[ n-k+1, k ]], {n, 1, rows}, {k, 1, n}]] (* _Jean-François Alcover, Jan 20 2012 *)

def A082137_row(n) : # as a triangular array
    var('z')
    s = (exp(z*x)/(1-tanh(x))).series(x,n+2)
    t = factorial(n)*s.coefficient(x,n)
    return [t.coefficient(z,n-k) for k in (0..n)]
for n in (0..7) : print(A082137_row(n))  # Peter Luschny, Aug 01 2012

Square array defined by T(n, k)=(2^(n-1)+0^n/2)C(n + k, n)= Sum{k=0..n, C(n+k, k+j)C(k+j, k)(1+(-1)^j)/2 }.
As an infinite lower triangular matrix, equals A007318 * A134309. - Gary W. Adamson, Oct 19 2007
O.g.f. for array read as a triangle: (1-x*(1+t))/((1-x)*(1-x*(1+2*t))) = 1 + x*(1+t) + x^2*(1+2*t+2*t^2) + x^3*(1+3*t+6*t^2+4*t^3) + .... - Peter Bala, Apr 26 2012
For array read as a triangle: T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) -2*T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 10 2013

A069727 Number of nonisomorphic unrooted Eulerian planar maps with n edges (Eulerian means that all vertices are of even valency; there is an Eulerian cycle).

Original entry on oeis.org

1, 1, 2, 4, 12, 34, 154, 675, 3534, 18985, 108070, 632109, 3807254, 23411290, 146734695, 934382820, 6034524474, 39457153432, 260855420489, 1741645762265, 11732357675908, 79673115468562, 545036528857605, 3753642607424647, 26010818244754788, 181266500331748878

Offset: 0

Valery A. Liskovets, Apr 07 2002

By duality, also the number of unrooted (sensed) bipartite maps with n edges. - Andrew Howroyd, Mar 29 2021

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
M. Deryagina, On the enumeration of hypermaps which are self-equivalent with respect to reversing the colors of vertices, Preprint 2016.
V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Cf. A000257 (rooted), A069720, A069724, A103939 (with distinguished face), A103940 (with distinguished vertex).

a[n_] := (1/(2n)) * (3*2^(n-1) * Binomial[2n, n]/((n+1)*(n+2)) + Sum[ EulerPhi[n/k] * d[n/k] * 2^(k-2) * Binomial[2k, k], {k, Most[ Divisors[n]]}]) + q[n]; a[0] = 1; q[n_?EvenQ] := 2^((n-4)/2)*Binomial[ n, n/2]/(n+2); q[n_?OddQ] := 2^((n-1)/2)*Binomial[(n-1), (n-1)/2]/(n+1); d[n_] := 4-Mod[n, 2]; Table[ a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 19 2011, after Valery A. Liskovets *)

a(n) = {if(n==0, 1, sumdiv(n, d, if(dAndrew Howroyd, Mar 29 2021

a(n) = (1/(2n))*(3*2^(n-1)*binomial(2n, n)/((n+1)(n+2)) + Sum_{k=1..n-1, k|n} phi(n/k)*d(n/k)*2^(k-2)*binomial(2k, k)) + q(n) where phi is the Euler function A000010, q(n) = 2^((n-4)/2)*binomial(n, n/2)/(n+2) if n is even, q(n) = 2^((n-1)/2)*binomial(n-1, (n-1)/2)/(n+1) if n is odd, d(n)=4, if n is even and d(n)=3 if n is odd. - Valery A. Liskovets, Mar 17 2005

a(n) ~ 3 * 2^(3*n-2) / (sqrt(Pi) * n^(7/2)). - Vaclav Kotesovec, Aug 28 2019

A183772 T(n,k) = 1/32 the number of (n+1) X (k+1) binary arrays with equal numbers of 2 X 2 subblocks with sum mod two being 0 and 1.

Original entry on oeis.org

0, 1, 1, 0, 6, 0, 12, 40, 40, 12, 0, 280, 0, 280, 0, 160, 2016, 7392, 7392, 2016, 160, 0, 14784, 0, 205920, 0, 14784, 0, 2240, 109824, 1555840, 5912192, 5912192, 1555840, 109824, 2240, 0, 823680, 0, 173065984, 0, 173065984, 0, 823680, 0, 32256, 6223360

Offset: 1

R. H. Hardin, Jan 06 2011

Table starts
.....0........1...........0..............12..................0
.....1........6..........40.............280...............2016
.....0.......40...........0............7392..................0
....12......280........7392..........205920............5912192
.....0.....2016...........0.........5912192..................0
...160....14784.....1555840.......173065984........19855042560
.....0...109824...........0......5134924800..................0
..2240...823680...346131968....153876579840.....70577422755840
.....0..6223360...........0...4646469273600..................0
.32256.47297536.79420170240.141154845511680.258888921984516096

			Some solutions for 4 X 3:
..0..1..0....1..1..0....1..0..0....0..1..1....1..1..1....0..0..0....0..0..1
..0..1..1....0..1..0....0..1..0....1..0..1....0..0..0....1..0..0....1..1..1
..0..1..1....1..1..0....1..0..0....1..0..0....1..0..1....1..0..1....0..0..1
..1..1..0....1..1..1....0..1..0....1..1..1....0..1..1....0..1..1....1..0..1

R. H. Hardin, Table of n, a(n) for n = 1..220

Column 1 is A098400(n/2-1).

Column 2 is A069720.

A069724 Number of nonisomorphic unrooted unicursal planar maps with n edges (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

Original entry on oeis.org

1, 2, 9, 38, 214, 1253, 7925, 51620, 346307, 2365886, 16421359, 115384738, 819276830, 5868540399, 42357643916, 307753571520, 2249048959624, 16520782751969, 121915128678131, 903391034923548, 6719098772562182

Offset: 1

Valery A. Liskovets, Apr 07 2002

G. C. Greubel, Table of n, a(n) for n = 1..1000
V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Cf. A069720, A069727, A005470.

a[n_] := 1/(2 n) DivisorSum[n, If[OddQ[n/#], EulerPhi[n/#] 2^(#-2) Binomial[2 #, #], 0]&] + If[OddQ[n], 2^((n-3)/2) Binomial[n-1, (n-1)/2], 2^((n-6)/2) Binomial[n, n/2]]; Array[a, 21] (* Jean-François Alcover, Sep 18 2016 *)

There is an easy formula.

a(n) ~ 8^(n-1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 28 2019

A154690 Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*binomial(n,k), 0 <= k <= n.

Original entry on oeis.org

2, 3, 3, 5, 8, 5, 9, 18, 18, 9, 17, 40, 48, 40, 17, 33, 90, 120, 120, 90, 33, 65, 204, 300, 320, 300, 204, 65, 129, 462, 756, 840, 840, 756, 462, 129, 257, 1040, 1904, 2240, 2240, 2240, 1904, 1040, 257, 513, 2322, 4752, 6048, 6048, 6048, 6048, 4752, 2322, 513

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jan 14 2009

From G. C. Greubel, Jan 18 2025: (Start)
A more general triangle of coefficients may be defined by T(n, k, p, q) = (p^(n-k)*q^k + p^k*q^(n-k))*A007318(n, k). When (p, q) = (2, 1) this sequence is obtained.
Some related triangles are:
  (p, q) = (1, 1) : 2*A007318(n,k).
  (p, q) = (2, 2) : 2*A038208(n,k).
  (p, q) = (3, 2) : A154692(n,k).
  (p, q) = (3, 3) : 2*A038221(n,k). (End)

			Triangle begins as:
     2;
     3,    3;
     5,    8,     5;
     9,   18,    18,     9;
    17,   40,    48,    40,    17;
    33,   90,   120,   120,    90,    33;
    65,  204,   300,   320,   300,   204,    65;
   129,  462,   756,   840,   840,   756,   462,   129;
   257, 1040,  1904,  2240,  2240,  2240,  1904,  1040,   257;
   513, 2322,  4752,  6048,  6048,  6048,  6048,  4752,  2322,  513;
  1025, 5140, 11700, 16320, 16800, 16128, 16800, 16320, 11700, 5140, 1025;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Lakhtakia, R. Messier, V. K. Varadan, and V. V. Varadan, Use of combinatorial algebra for diffusion on fractals, Physical Review A, volume 34, Number 3 (1986) p. 2502, Fig. 3.

Cf. A000129, A001045, A025192, A038208, A038221, A069720, A107920, A154692.
Cf. A215149.
Sums include: A008776 (row), A010673 (alternating sign row).
Columns k: A000051 (k=0).
Main diagonal: A059304.

A154690:= func< n,k | (2^(n-k)+2^k)*Binomial(n,k) >;
[A154690(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 18 2025

A154690 := proc(n,m) binomial(n,m)*(2^(n-m)+2^m) ; end proc: # R. J. Mathar, Jan 13 2011

T[n_, m_]:= (2^(n-m) + 2^m)*Binomial[n,m];
Table[T[n,m], {n,0,12}, {m,0,n}]//Flatten

from sage.all import *
def A154690(n,k): return (pow(2,n-k)+pow(2,k))*binomial(n,k)
print(flatten([[A154690(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 18 2025

T(n, k) = (2^(n-k) + 2^k)*A007318(n, k).
Sum_{k=0..n} T(n, k) = A008776(n) = A025192(n+1).
From G. C. Greubel, Jan 18 2025: (Start)
T(n, n-k) = T(n, k) (symmetry).
T(n, 1) = n + A215149(n), n >= 1.
T(2*n-1, n) = 3*A069720(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A010673(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000129(n+1) + A001045(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = n+1 + A107920(n+1). (End)

A082143 First subdiagonal of number array A082137.

Original entry on oeis.org

1, 3, 20, 140, 1008, 7392, 54912, 411840, 3111680, 23648768, 180590592, 1384527872, 10650214400, 82158796800, 635361361920, 4924050554880, 38233804308480, 297374033510400, 2316387208396800, 18067820225495040, 141101072237199360, 1103153837490831360

Offset: 0

Paul Barry, Apr 06 2003

			a(0)=(2^(-1)+(0^0)/2)C(1,0)=2*(1/2)=1 (use 0^0=1).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A069723, A082144, A082145.

Cf. A000079, A001700, A069720.

a082143 0 = 1
a082143 n = (a000079 $ n - 1) * (a001700 n)
-- Reinhard Zumkeller, Jan 15 2015

[(2^(n-1) + 0^n/2)*Binomial(2*n+1,n): n in [0..30]]; // G. C. Greubel, Feb 05 2018

Join[{1}, Table[2^(n-1)* Binomial[2*n+1,n], {n,1,30}]] (* G. C. Greubel, Feb 05 2018 *)

for(n=0,30, print1((2^(n-1) + 0^n/2)*Binomial(2*n+1,n), ", ")) \\ G. C. Greubel, Feb 05 2018

a(n) = (2^(n-1) + 0^n/2)*C(2n+1, n).
Conjecture: (n+1)*a(n) +4*(-2*n-1)*a(n-1)=0. - R. J. Mathar, Oct 19 2014
From Reinhard Zumkeller, Jan 15 2015: (Start)
a(n) = A000079(n-1) * A001700(n), for n > 0.
a(n) = A069720(n+1)/2. (End)
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 64*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)) + 1/7.
Sum_{n>=0} (-1)^n/a(n) = 32*log(2)/27 - 1/9. (End)

A098400 a(n) = 4^nbinomial(2n+1, n).

Original entry on oeis.org

1, 12, 160, 2240, 32256, 473088, 7028736, 105431040, 1593180160, 24216338432, 369849532416, 5671026163712, 87246556364800, 1346089726771200, 20819521107394560, 322702577164615680, 5011381198321090560, 77954818640550297600, 1214454016715941478400

Offset: 0

Paul Barry, Sep 06 2004

G. C. Greubel, Table of n, a(n) for n = 0..820

Cf. A000108, A001700, A069720, A069723, A098399, A151403.

[4^n*(2*n+1)*Catalan(n): n in [0..30]]; // G. C. Greubel, Dec 27 2023

Table[4^n Binomial[2n+1,n],{n,0,20}] (* Harvey P. Dale, Jan 22 2019 *)

a(n)=binomial(2*n+1,n)<<(2*n) \\ Charles R Greathouse IV, Oct 23 2023

[4^n*binomial(2*n+1,n) for n in range(31)] # G. C. Greubel, Dec 27 2023

G.f.: (1-sqrt(1-16*x))/(8*x*sqrt(1-16*x)).
E.g.f.: a(n) = n! * [x^n] exp(8*x)*(BesselI(0, 8*x) + BesselI(1, 8*x)). - Peter Luschny, Aug 25 2012
(n+1)*a(n) - 8*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 26 2012
a(n) = 4^n*(2*n+1)*Hypergeometric2F1([1-n,-n],[2],1). - Peter Luschny, Sep 22 2014
From G. C. Greubel, Dec 27 2023: (Start)
a(n) = 4^n * A001700(n).
a(n) = 4^n * (2*n+1) * A000108(n).
a(n) = (2*n+1) * A151403(n). (End)
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 8/15 + 128*arcsin(1/4)/(15*sqrt(15)).
Sum_{n>=0} (-1)^n/a(n) = 8/17 + 128*arcsinh(1/4)/(17*sqrt(17)). (End)

A069722 Number of rooted unicursal planar maps with n edges and exactly one vertex of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

Original entry on oeis.org

0, 4, 24, 160, 1120, 8064, 59136, 439296, 3294720, 24893440, 189190144, 1444724736, 11076222976, 85201715200, 657270374400, 5082890895360, 39392404439040, 305870434467840, 2378992268083200, 18531097667174400, 144542561803960320, 1128808577897594880

Offset: 1

Valery A. Liskovets, Apr 07 2002

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Valery A. Liskovets and Timothy R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Cf. A059304, A069720, A069721, A069723, A089156.

[0] cat[2^(n-1)*Binomial(2*n-2, n-1): n in [2..20]]; // Vincenzo Librandi, Nov 17 2011

Z:=(1-sqrt(1-z))*8^n/sqrt(1-z): Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=0..19); # Zerinvary Lajos, Jan 01 2007

Join[{0},Table[2^(n-1) Binomial[2n-2,n-1],{n,2,20}]] (* Harvey P. Dale, Nov 16 2011 *)

a(n) = 2^(n-1)*binomial(2n-2, n-1), n>1.
a(n) = 2*A069723(n), n>1.
G.f. for a(n)^2: 1/AGM(1, (1-64*x)^(1/2)). - Benoit Cloitre, Jan 01 2004
a(n) = A059304(n-1), n>1. [R. J. Mathar, Sep 29 2008]
a(n) ~ 2^(3*n-3)/sqrt(Pi*n). - Vaclav Kotesovec, Sep 28 2019
E.g.f.: x * (exp(4*x) * (BesselI(0,4*x) - BesselI(1,4*x)) - 1). - Ilya Gutkovskiy, Nov 03 2021
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=2} 1/a(n) = 1/7 + 8*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).
Sum_{n>=2} (-1)^n/a(n) = 1/9 + 4*log(2)/27. (End)

A098399 a(n) = 3^nbinomial(2n+1, n).

Original entry on oeis.org

1, 9, 90, 945, 10206, 112266, 1250964, 14073345, 159497910, 1818276174, 20827527084, 239516561466, 2763652632300, 31979409030900, 370961144758440, 4312423307816865, 50227047938102310, 585982225944526950, 6846739692614999100, 80106854403595489470, 938394580156404305220

Offset: 0

Paul Barry, Sep 06 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Cf. A000108, A001700, A005159, A069720, A069723, A098400.

[3^n*Binomial(2*n+1, n): n in [ 0..20]]; // Vincenzo Librandi, Nov 24 2012

Z:=(1-sqrt(1-3*z))*4^n/sqrt(1-3*z)/6: Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=1..18); # Zerinvary Lajos, Jan 01 2007

Table[3^n Binomial[2n+1,n], {n,0,20}] (* Harvey P. Dale, Mar 28 2012 *)

a(n)=binomial(2*n+1,n)*3^n \\ Charles R Greathouse IV, Oct 23 2023

[3^n*binomial(2*n+1, n) for n in range(21)] # G. C. Greubel, Dec 27 2023

G.f.: (1-sqrt(1-12*x))/(6*x*sqrt(1-12*x)).
E.g.f.: a(n) = n!* [x^n] exp(6*x)*(BesselI(0, 6*x) + BesselI(1, 6*x)). - Peter Luschny, Aug 25 2012
(n+1)*a(n) - 6*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 24 2012
From G. C. Greubel, Dec 27 2023: (Start)
a(n) = 3^n * (2*n+1)*A000108(n).
a(n) = (2*n+1)*A005159(n).
a(n) = 3^n * A001700(n). (End)
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 6/11 + 72*arcsin(1/(2*sqrt(3)))/(11*sqrt(11)).
Sum_{n>=0} (-1)^n/a(n) = 6/13 + 72*arcsinh(1/(2*sqrt(3)))/(13*sqrt(13)). (End)

cp's OEIS Frontend

A069723 a(n) = 2^(n-1)*binomial(2*n-3, n-1).

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A082137 Square array of transforms of binomial coefficients, read by antidiagonals.

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A069727 Number of nonisomorphic unrooted Eulerian planar maps with n edges (Eulerian means that all vertices are of even valency; there is an Eulerian cycle).

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A183772 T(n,k) = 1/32 the number of (n+1) X (k+1) binary arrays with equal numbers of 2 X 2 subblocks with sum mod two being 0 and 1.

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A069724 Number of nonisomorphic unrooted unicursal planar maps with n edges (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

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A154690 Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*binomial(n,k), 0 <= k <= n.

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A082143 First subdiagonal of number array A082137.

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A069723 a(n) = 2^(n-1)binomial(2n-3, n-1).

A098400 a(n) = 4^nbinomial(2n+1, n).

A098399 a(n) = 3^nbinomial(2n+1, n).