A000257 -id:A000257 - cp's OEIS frontend

A186266 Expansion of 2F1( 1/2, 3/2; 4; 16*x ).

1, 3, 18, 140, 1260, 12474, 132132, 1472328, 17065620, 204155380, 2506399896, 31443925968, 401783498480, 5215458874500, 68633685693000, 914099013896400, 12304253831789700, 167193096184907100, 2291164651422801000, 31637804708163654000, 439903041116118980400

Offset: 0

Olivier Gérard, Feb 16 2011

Combinatorial interpretation welcome.

Could involve planar maps, lattice walks, and interpretations of Catalan numbers.

Indranil Ghosh, Table of n, a(n) for n = 0..800
H. Franzen and T. Weist, The Value of the Kac Polynomial at One, arXiv preprint arXiv:1608.03419 [math.RT], 2016.

Formula close to A000257, A000888, A172392.

Cf. A000108.

CoefficientList[
Series[HypergeometricPFQ[{1/2, 3/2}, {4}, 16*x], {x, 0, 20}], x]
Table[3 CatalanNumber[n] CatalanNumber[n+1] * (n+1) / (n+3), {n, 0, 20}] (* Indranil Ghosh, Mar 05 2017 *)

c(n) = binomial(2*n,n) / (n+1);
a(n) = 3 * c(n) * c(n+1) *(n+1) / (n+3); \\ Indranil Ghosh, Mar 05 2017

import math
f=math.factorial
def C(n,r): return f(n) / f(r) / f(n-r)
def Catalan(n): return C(2*n, n) / (n+1)
def A186266(n): return 3 * Catalan(n) * Catalan(n+1) * (n+1) / (n+3) # Indranil Ghosh, Mar 05 2017

a(n) = 3*A000108(n)*A000108(n+1)*(n+1)/(n+3). - David Scambler, Aug 18 2012

D-finite with recurrence n*(n+3)*a(n) -4*(2*n-1)*(2*n+1)*a(n-1)=0. - R. J. Mathar, Jun 17 2016

A275607 a(n) = 212^nGamma(n+1/2)(n+1)/(sqrt(Pi)Gamma(n+3)).

Original entry on oeis.org

1, 4, 27, 216, 1890, 17496, 168399, 1667952, 16888014, 173997720, 1818276174, 19225409616, 205299909828, 2210922105840, 23984556773175, 261854925711840, 2874948871877910, 31722346066169880, 351589335566716170, 3912422681494285200, 43694647856506630620, 489597172255515289680

Offset: 0

Karol A. Penson, Nov 14 2016

nonn

In reference of K. Szymanski et al. the function g(x) from the Eq.(4.6) satisfies the equality g(x/4)/4 = W(x) where W(x) is the weight function of the integral representation, see below.

G. C. Greubel, Table of n, a(n) for n = 0..925
Simeon T. Stefanov, Counting fixed points free vector fields on B^2, arXiv:1807.03714 [math.GT], 2018.
K. Szymanski, B. Collins, T. Szarek and K. Zyczkowski, Convex set of quantum states with positive partial transpose analysed by hit and run algorithm, arXiv:1611.01194 [quant-ph], 2016.

Cf. A000108, A002293, A002894, A000168, A000139, A000257, A004987, A005568, A000888, A004981.

a := n -> (2^(2*n+1)*3^n*(n+1)*GAMMA(n+1/2))/(sqrt(Pi)*GAMMA(n+3)):
seq(a(n), n=0..21); # Peter Luschny, Nov 14 2016

g[z_] :=  E^z (BesselI[0,z] - (1-1/z) BesselI[1,z])
Table[CoefficientList[2/3 Series[g[6z], {z,0,21}],z]] Range[0, 21]! //Flatten (* Peter Luschny, Nov 14 2016 *)
Table[ 2*12^n*(n + 1)*Gamma[n + 1/2]/(Sqrt[Pi]*Gamma[n + 3]), {n,0,100}] (* G. C. Greubel, Jan 13 2017 *)

a(n)=2*12^n*gamma(n+1/2)*(n+1)\/(sqrt(Pi)*(n+2)!) \\ Charles R Greathouse IV, Nov 14 2016

a(n)=2*3^n*binomial(2*n+1,n-1)*(n+1)/(2*n+1)/n \\ Charles R Greathouse IV, Nov 14 2016

O.g.f: (1/54)*(1-(6*z+1)*sqrt(1-12*z))/z^2;
E.g.f.(in Maple notation): (1/9)*exp(6*z)*(6*z*(BesselI(0,6*z)-BesselI(1,6*z))+ BesselI(1,6*z))/z;
Recurrence: (-12*n^2-54*n-54)*a(n+1)+(n^2+6*n+8)*a(n+2)=0, n=0,1..., for the initial values a(0)=1, a(1)=4.
Integral representation as the n-th Hausdorff moment of the positive function W(x) on the segment x=(0,12), i.e., a(n) = Integral_{x=0..12} x^n*W(x) dx, where W(x) = (1/27)*sqrt(12-x)*(3+(1/2)*x)/(Pi*sqrt(x)). This representation is unique.
a(n) ~ 2^(2*n+1)*3^n/(sqrt(Pi)*n^(3/2)). - Ilya Gutkovskiy, Nov 14 2016
a(n) = 2*3^n*binomial(2n+1, n-1)*(n+1)/(2n^2+n). - Charles R Greathouse IV, Nov 14 2016

A331836 Number of noncrossing anti-commutator friendly partitions on {1,2,...,2n}.

Original entry on oeis.org

1, 5, 22, 117, 678, 4162, 26588, 174925, 1177158, 8064854, 56062804, 394443458, 2803490524, 20098913252, 145175116408, 1055463627197, 7717664983366, 56720231324046, 418757618733092, 3104269959560566

Offset: 1

Kamil Szpojankowski, Jan 28 2020

Let n be a positive integer, let sigma be a noncrossing partition on {1,2,...,2n}. Consider the set OuterMax(sigma) := {max(W); W is an outer block of sigma}.
We say that sigma is anti-commutator friendly when it satisfies:
1) OuterMax(sigma) is a subset of the union of {1,3,...,2n-1} and {2n}.
2) For every j in {1, 3,...,2n-1} \ OuterMax(sigma), one has depth(j) is not equal to depth(j + 1), where depth(j) stands for the depth of the block of sigma which contains the number j.
For example for n=1 only partition {{1},{2}} is anti-commutator friendly.
Multiplied by two, gives sequence of Boolean cumulants of ab+ba, for a,b freely independent both distributed (delta_0+delta_2)/2. See M. Fevrier et al. Proposition 6.11.

M. Fevrier, M. Mastnak, A. Nica and K. Szpojankowski, Using Boolean cumulants to study multiplication and anti-commutators of free random variables, arXiv:1907.10842 [math.OA], 2019.

Cf. A000257.

Rest[CoefficientList[Series[1/2 - Sqrt[(1 - 8*x)*(1 - Sqrt[1 - 8*x] - 2*x)/(8*x)], {x, 0, 20}], x]] (* Vaclav Kotesovec, Jan 29 2020 *)

seq(n)={Vec(1/2 - sqrt((1 - 8*x)*(1-2*x-sqrt(1 - 8*x + O(x^2*x^n)))/(8*x)))} \\ Andrew Howroyd, Jan 28 2020

G.f.: 1/2 - sqrt( (1 - 8*z)*(1 - 2*z-sqrt(1-8*z))/(8*z) ).
G.f.: 1/2 - sqrt( 1/4 + 3*z - 4*z*g(z) ), where g(z) is the g.f. of A000257.
a(n) ~ sqrt(3) * 2^(3*n-2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 29 2020
D-finite with recurrence n*(2*n+1)*a(n) +2*(-11*n^2+17*n-3)*a(n-1) +24*(-n^2+7*n-8)*a(n-2) +32*(16*n^2-109*n+186)*a(n-3) +256*(n-4)*(2*n-9)*a(n-4)=0. - R. J. Mathar, Mar 06 2022

A361139 Number of rooted bipartite maps of genus 1/2 with n edges.

Original entry on oeis.org

0, 1, 9, 69, 510, 3738, 27405, 201569, 1488762, 11043318, 82257890, 615092178, 4615882908, 34752865332, 262437282621, 1987229885913

Offset: 1

R. J. Mathar, Mar 02 2023

Valentin Bonzom, Guillaume Chapuy, and Maciej Dolega, Enumeration of non-oriented maps via integrability, Alg. Combin. 5 (6) (2022) pp. 1363-1390, A.2.

Cf. A000257 (genus 0).

A361140 Number of rooted bipartite maps of genus 1 with n edges.

Original entry on oeis.org

0, 0, 4, 63, 720, 7254, 68460, 621315, 5496208, 47759130, 409620156, 3478672642, 29315742924, 245539064736, 2046309441924, 16983591315267

Offset: 1

R. J. Mathar, Mar 02 2023

Valentin Bonzom, Guillaume Chapuy, and Maciej Dolega, Enumeration of non-oriented maps via integrability, Alg. Combin. 5 (6) (2022) pp. 1363-1390, A.2.

Cf. A000257 (genus 0).

cp's OEIS Frontend

A186266 Expansion of 2F1( 1/2, 3/2; 4; 16*x ).

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A275607 a(n) = 2*12^n*Gamma(n+1/2)*(n+1)/(sqrt(Pi)*Gamma(n+3)).

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A331836 Number of noncrossing anti-commutator friendly partitions on {1,2,...,2n}.

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A361139 Number of rooted bipartite maps of genus 1/2 with n edges.

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A361140 Number of rooted bipartite maps of genus 1 with n edges.

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A275607 a(n) = 212^nGamma(n+1/2)(n+1)/(sqrt(Pi)Gamma(n+3)).