A135404 -id:A135404 - cp's OEIS frontend

A000531 From area of cyclic polygon of 2n + 1 sides.

1, 7, 38, 187, 874, 3958, 17548, 76627, 330818, 1415650, 6015316, 25413342, 106853668, 447472972, 1867450648, 7770342787, 32248174258, 133530264682, 551793690628, 2276098026922, 9373521044908, 38546133661492

Offset: 1

Simon Plouffe

Expected number of matches remaining in Banach's original matchbox problem (counted when empty box is chosen), multiplied by 2^(2*n-1). - Michael Steyer, Apr 13 2001

A conjectured definition: Let 0 < a_1 < a_2 <...

a(n) = total weight of upsteps in all Dyck n-paths (A000108) when each upstep is weighted with its position in the path. For example, the Dyck path UDUUDUDD has upsteps in positions 1,3,4,6 and contributes 1+3+4+6=14 to the weight for Dyck 4-paths. The summand (n-k)*binomial(2*n+1, k) in the Maple formula below is the total weight of upsteps terminating at height n-k, 0<=k<=n-1. - David Callan, Dec 29 2006

Catalan transform of binomial transform of squares. - Philippe Deléham, Oct 31 2008

a(n) is also the number of walks of length 2n in the quarter plane starting and ending at the origin using steps {(1,1),(1,0),(-1,0), (-1,-1)} (which appear in Gessel's conjecture) in which the steps (1,0) and (-1,0) appear exactly once each. - Arvind Ayyer, Mar 02 2009

Equals the Catalan sequence, A000108, convolved with A002457: (1, 6, 30, 140, ...). - Gary W. Adamson, May 14 2009

Total number of occurrences of the pattern 213 (or 132) in all skew-indecomposable (n+2)-permutations avoiding the pattern 123. For example, a(1) = 1, since there is one occurrence of the pattern 213 in the set {213, 132}. - Cheyne Homberger, Mar 13 2013

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I.

T. D. Noe, Table of n, a(n) for n = 1..100
Arvind Ayyer, Towards a human proof of Gessel's conjecture, arXiv:0902.2329 [math.CO], 2009, JIS 12 (2009) 09.4.2
Hacène Belbachir, Toufik Djellal, Jean-Gabriel Luque, On the self-convolution of generalized Fibonacci numbers, arXiv:1703.00323 [math.CO], 2017.
F. Bowman, Cyclic pentagons, Math. Gaz. 36, (1952). 244-250. MR0051523.
A. Burstein and S. Elizalde, Total occurrence statistics on restricted permutations, arXiv preprint arXiv:1305.3177 [math.CO], 2013.
C. Homberger, Expected patterns in permutation classes, Electronic Journal of Combinatorics, 19(3) (2012), P43.
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.
Yasuhiko Kamiyama, The Euler characteristic of the regular spherical polygon spaces, arXiv:1803.05559 [math.GT], 2018.
A. F. Moebius, Über die Gleichungen, mittelst welcher aus den Seiten eines in einen Kreis zu beschreibenden Vielecks der Halbmesser des Kreises und die Fläche des Vielecks gefunden werden, Gesammelte Werke, vol. 1., pp. 407-438.
D. P. Robbins, Areas of polygons inscribed in a circle, Amer. Math. Monthly, 102 (1995), 523-530.
Eric Weisstein's World of Mathematics, Cyclic Polygon
Y. Q. Zhao, Introduction to Probability with Applications

Cf. A002457 (Banach's modified matchbox problem), A135404, A002457, A258431.

f := proc(n) sum((n-k)*binomial(2*n+1,k),k=0..n-1); end;

a[n_] := ((2n+1)!/n!^2-4^n)/2; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Dec 07 2011, after Pari *)

PARI
```
a(n)=if(n<1,0,((2*n+1)!/n!^2-4^n)/2)
```

a(n) = ((2n+1)!/((n!)^2)-4^n)/2. - Simon Norton (simon(AT)dpmms.cam.ac.uk), May 14 2001
na(n) = (8n-2)a(n-1) - (16n-8)a(n-2), n>1. - Michael Somos, Apr 18 2003
E.g.f.: 1/2*((1+4*x)*exp(2*x)*BesselI(0, 2*x) + 4*x*exp(2*x)*BesselI(1, 2*x) - exp(4*x)). - Vladeta Jovovic, Sep 22 2003
a(n-1) = 4^n*sum_{k=0..n} binomial(2*k+1, k)*4^(-k) = (2*n+1)*(2*n+3)*C(n) - 2^(2*n+1) (C(n) = Catalan); g.f.: x*c(x)/(1-4*x)^(3/2), c(x): g.f. of Catalan numbers A000108. - Wolfdieter Lang
a(n) = Sum_{k=0..n} A039599(n,k)*k^2, for n>=1. - Philippe Deléham, Jun 10 2007
a(n) = Sum_{k=0..n} A106566(n,k)*A001788(k). - Philippe Deléham, Oct 31 2008
(Conjecture) a(n)=2^(2*n)*sum_{k=1..n} cos(k*Pi/(2*n+1))^2*n. - L. Edson Jeffery, Jan 21 2012

Moebius reference from Michael Somos

A186229 Expansion of (2F1( (-(1/2), 1/6); (-2/3))( 16 x) -1)/(2*x).

Original entry on oeis.org

1, 14, 182, 2470, 34580, 494760, 7191690, 105793545, 1570873850, 23500272796, 353724885332, 5351515200668, 81313973049064, 1240116577389200, 18973783634054760, 291115203548084370, 4477664537437798980, 69023046543088792440, 1066084706728274263800, 16495237916832025427160, 255635559046076610807120

Offset: 0

Olivier Gérard, Feb 15 2011

Combinatorial interpretation welcome.

Probably a class of paths (Cf. A135404, A000888)

Vincenzo Librandi, Table of n, a(n) for n = 0..200

CoefficientList[Series[(HypergeometricPFQ[{-(1/2), 1/6}, {-(2/3)}, 16 x] - 1)/(2 x), {x, 0, 20}], x]
FullSimplify[Table[-((2^(1/3 + 4 n) (-(4/3))! (-(1/2) + n)! (1/6 + n)!)/(Pi (-(2/3) + n)! (1 + n)!)), {n, 0, 20}]] (* Benedict W. J. Irwin, Jul 12 2016 *)

D-finite with recurrence (n+1)*(3n-2)*a(n) = 4*(6n+1)*(2n-1)*a(n-1). - R. J. Mathar, Jul 11 2012
a(n) ~ 3*GAMMA(2/3)*2^(1/3) * 16^n/(Pi*n^(2/3)). - Vaclav Kotesovec, Aug 13 2013
a(n) = -2^(1/3+4*n)*(-4/3)!*(-1/2+n)!*(1/6+n)!/(Pi*(-2/3+n)!*(1+n)!). - Benedict W. J. Irwin, Jul 12 2016

A186231 Expansion of ( 2F1([-1/4, 1/4]; [-1/2], 16x) - 1 ) / (2x).

Original entry on oeis.org

1, 15, 210, 3003, 43758, 646646, 9657700, 145422675, 2203961430, 33578000610, 513791607420, 7890371113950, 121548660036300, 1877405874732108, 29065024282889672, 450883717216034179, 7007092303604022630, 109069992321755544170, 1700179760011004467468, 26536589497469056215210, 414670662257153823494820

Offset: 0

Olivier Gérard, Feb 15 2011

nonn

Combinatorial interpretation welcome.
Probably a class of paths (Cf. A135404, A000888).
Number of North-East lattice paths from (0,0) to (n,n+1). - Michael D. Weiner, Apr 14 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A186229.

CoefficientList[Series[(HypergeometricPFQ[{-(1/4), 1/4}, {-(1/2)}, 16 x] - 1)/(2 x), {x, 0, 20}], x]

a(n) = A001791(2n+1). - R. J. Mathar, Jul 10 2012

D-finite with recurrence -(n+1)*(2*n-1)*a(n) +2*(4*n-1)*(4*n+1)*a(n-1)=0. - R. J. Mathar, Apr 26 2017

A157513 Triangle of numbers of walks in the quarter-plane, of length 2n beginning and ending at the origin using steps {(1,1), (1,0), (-1,0), (-1,-1)} (Gessel steps) arranged according to the number of times the steps (1,1) and (-1,-1) occur.

Original entry on oeis.org

1, 1, 1, 2, 7, 2, 5, 37, 38, 5, 14, 177, 390, 187, 14, 42, 806, 3065, 3175, 874, 42, 132, 3566, 20742, 37260, 22254, 3958, 132, 429, 15485, 127575, 351821, 365433, 141442, 17548, 429, 1430, 66373, 734332, 2876886, 4597444, 3100670, 839068, 76627, 1430

Offset: 0

Arvind Ayyer, Mar 02 2009

The first and the last terms in each row are Catalan numbers. The sum in each row gives the Gessel sequence.

			For n=2, there are 2 walks of length 4 where the diagonal steps (1,1) and (-1,-1) occur zero times [(1,0),(1,0),(-1,0),(-1,0)] and [(1,0),(-1,0),(1,0),(-1,0)];
7 walks where the diagonal steps occur once [(1,0),(-1,0),(1,1),(-1,-1)], [(1,1),(-1,-1),(1,0),(-1,0)],  [(1,0),(1,1),(-1,0),(-1,-1)],  [(1,0),(1,1),(-1,-1),(-1,0)],  [(1,1),(1,0),(-1,0),(-1,-1)],  [(1,1),(1,0),(-1,-1),(-1,0)],  [(1,1),(-1,0),(1,0),(-1,-1)];
and finally 2 walks where the diagonal steps occur twice [(1,1),(1,1),(-1,-1),(-1,-1)] and [(1,1),(-1,-1),(1,1),(-1,-1)].
Triangle begins:
1;
1,     1;
2,     7,    2;
5,    37,   38,    5;
14,  177,  390,  187,   14;
42,  806, 3065, 3175,  874,  42;

Alois P. Heinz, Table of n, a(n) for n = 0..5049
Arvind Ayyer, Towards a human proof of Gessel's conjecture, arXiv:0902.2329 [math.CO], 2009.
Manuel Kauers, Christoph Koutschan and Doron Zeilberger, Proof of Ira Gessel's Lattice Path Conjecture
Marko Petkovsek and Herbert S. Wilf, On a conjecture of Ira Gessel, arXiv:0807.3202 [math.CO], 2008.

Cf. A135404, A000531, A000108.

b:= proc(n, k, t, x, y) option remember; `if` (min(n, x, y, k, t, n-x)<0, 0, `if` (n=0, `if` (max(n, k, t)=0, 1, 0), b(n-1, k-1, t, x+1, y+1) +b(n-1, k, t, x+1, y) +b(n-1, k, t, x-1, y) +b(n-1, k, t-1, x-1, y-1))) end: T:= (n,k)-> b(2*n, k, k, 0, 0):
seq (seq (T(n, k), k=0..n), n=0..8);  # Alois P. Heinz, Jul 04 2011

b[n_, k_, t_, x_, y_] := b[n, k, t, x, y] = If[Min[n, x, y, k, t, n-x] < 0, 0, If[n == 0, If[Max[n, k, t] == 0, 1, 0], b[n-1, k-1, t, x+1, y+1] + b[n - 1, k, t, x+1, y] + b[n-1, k, t, x-1, y] + b[n-1, k, t-1, x-1, y-1]]]; T[n_, k_] := b[2*n, k, k, 0, 0]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 27 2016, after Alois P. Heinz *)

A292361 The number of paths of length 2m in the plane, starting and ending at (0,1), with unit steps in the four directions (north, east, south, west) and staying in the region y > 0 or x > -y.

Original entry on oeis.org

1, 3, 21, 192, 2009, 22818, 273895, 3421318, 44042729, 580473551, 7796745921, 106365396629, 1470068855112, 20543335134692, 289818595800636, 4122517765350669, 59066177091706608

Offset: 0

Timothy Budd, Sep 15 2017

T. Budd, Winding of simple walks on the square lattice, arXiv:1709.04042 [math.CO], 2017.

Cf. A135404.

a[n_] := SeriesCoefficient[-Pi(1 + 2 Sum[(y+3y^2+y^3)/(1+y+y^2+y^3+y^4) /. y->EllipticNomeQ[m]^l, {l,n+1}])/(4EllipticK[m]) /. m->16x, {x,0,n+1}]

G.f.: A(x) = 1/(2x) - (Pi / (4 x K(16x))) * (1 + 2 Sum_{n>=1} (q^n + 3q^(2n)+ q^(3n)) / (1 + q^n + q^(2n) + q^(3n) + q^(4n)) ), where q=q(16x) is the Jacobi nome of parameter m=16x and K(16x) is the complete elliptic integral of the first kind of parameter m=16x (proven).

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A000531 From area of cyclic polygon of 2n + 1 sides.

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A186229 Expansion of (2F1( (-(1/2), 1/6); (-2/3))( 16 x) -1)/(2*x).

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A186231 Expansion of ( 2F1([-1/4, 1/4]; [-1/2], 16*x) - 1 ) / (2*x).

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A157513 Triangle of numbers of walks in the quarter-plane, of length 2n beginning and ending at the origin using steps {(1,1), (1,0), (-1,0), (-1,-1)} (Gessel steps) arranged according to the number of times the steps (1,1) and (-1,-1) occur.

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Programs

Maple

Mathematica

A292361 The number of paths of length 2m in the plane, starting and ending at (0,1), with unit steps in the four directions (north, east, south, west) and staying in the region y > 0 or x > -y.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A186231 Expansion of ( 2F1([-1/4, 1/4]; [-1/2], 16x) - 1 ) / (2x).