A052718 -id:A052718 - cp's OEIS frontend

A052712 Expansion of e.g.f. (1 + 4x - sqrt(1-8x))/8.

0, 1, 2, 24, 480, 13440, 483840, 21288960, 1107025920, 66421555200, 4516665753600, 343266597273600, 28834394170982400, 2652764263730380800, 265276426373038080000, 28649854048288112640000

Offset: 0

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

Also the number of random walk labelings of the 2 X (n-1) king's graph, for n > 1. - Sela Fried, Apr 14 2023

G. C. Greubel, Table of n, a(n) for n = 0..330
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 668
Sela Fried and Toufik Mansour, Graph labelings obtainable by random walks, arXiv:2304.05728 [math.CO], 2023.

Cf. A052711, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052721, A052722, A052723.

Cf. A000108, A003645.

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

Table[n!*2^(n-2)*CatalanNumber[n-1] +Boole[n==1]/2 +Boole[n==0]/4, {n,0,30}] (* G. C. Greubel, May 30 2022 *)

[2^(n-2)*factorial(n)*catalan_number(n-1) +bool(n==0)/8 +bool(n==1)/2 for n in (0..30)] # G. C. Greubel, May 30 2022

D-finite with recurrence: a(0) = 0, a(1)=1, a(2)=2, a(n+1) = 4*(2*n-1)*a(n).
a(n) = 8^(n+1)*Gamma(n+3/2)/sqrt(Pi).
a(n) = n!*A003645(n-2), n>1. - R. J. Mathar, Oct 18 2013
G.f.: (1 + 4*x - 2F0([1,-1/2], [], 8*x))/8. - R. J. Mathar, Jan 25 2020

A052723 Expansion of e.g.f. (1 - x - sqrt(1-2x+x^2-4x^3))/(2*x).

Original entry on oeis.org

0, 0, 2, 6, 24, 240, 2880, 35280, 524160, 9434880, 188697600, 4151347200, 101548339200, 2727435110400, 79332244992000, 2488504322304000, 83879464660992000, 3021209014247424000, 115754916599562240000

Offset: 0

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

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 679

Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052721, A052722.

Cf. A023431.

spec := [S,{B=Prod(S,S),C=Union(B,S,Z),S=Prod(Z,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(n!*add(binomial(n-2-k,2*k)*binomial(2*k,k)/(k+1), k=0..floor((n-2)/3)), n=0..18);  # Mark van Hoeij, May 12 2013

With[{nn=20},CoefficientList[Series[(1-x-Sqrt[1-2x+x^2-4x^3])/(2x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 19 2017 *)
a[n_]:= a[n]= n!*Sum[Binomial[n-k-2,2*k]*CatalanNumber[k], {k,0,Floor[(n-2)/2]}];
Table[a[n], {n,0,30}] (* G. C. Greubel, May 28 2022 *)

def A052723(n): return factorial(n)*sum( binomial(n-k-2, 2*k)*catalan_number(k) for k in (0..(n-2)//2) )
[A052723(n) for n in (0..30)] # G. C. Greubel, May 28 2022

D-finite with recurrence: a(0) = a(1) = 0, a(2) = 2, a(3) = 6, a(4) = 24, (n+4)*a(n+3) = (15 + 11*n + 2*n^2)*a(n+2) - (6 + 11*n + 6*n^2 + n^3)*a(n+1) - (12 - 2*n - 32*n^2 - 22*n^2 - 4*n^4)*a(n).

a(n) = n!*A023431(n-2). - R. J. Mathar, Oct 18 2013

A052711 Expansion of e.g.f. x(1 - 2x - sqrt(1-4*x))/2.

Original entry on oeis.org

0, 0, 0, 6, 48, 600, 10080, 211680, 5322240, 155675520, 5189184000, 194075481600, 8045310873600, 366061644748800, 18134130709094400, 971471287987200000, 55956746188062720000, 3448334483839365120000

Offset: 0

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

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 667

Cf. A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052721, A052722, A052723.

Cf. A000108, A052717.

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

With[{nn=20},CoefficientList[Series[x (1-2x-Sqrt[1-4x])/2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 05 2016 *)
Table[n!*CatalanNumber[n-2] +Boole[n==1] -2*Boole[n==2], {n,0,30}] (* G. C. Greubel, May 30 2022 *)

[factorial(n)*catalan_number(n-2) + bool(n==1)/2 - 2*bool(n==2) for n in (0..30)] # G. C. Greubel, May 30 2022

D-finite with recurrence: a(1)=0, a(2)=0, a(3)=6, a(4)=48, n*a(n+1) = 2*(n+1)*(2*n-3)*a(n).
From R. J. Mathar, Oct 18 2013: (Start)
a(n) = n!*A000108(n-2).
a(n) = A052717(n), n>2. (End)
G.f.: x*(1 - 4*x - 2F0([-1/2,2], [], 4*x))/2. - R. J. Mathar, Jan 25 2020

A052713 Expansion of e.g.f. (1-sqrt(1-8*x))/2.

Original entry on oeis.org

0, 2, 8, 96, 1920, 53760, 1935360, 85155840, 4428103680, 265686220800, 18066663014400, 1373066389094400, 115337576683929600, 10611057054921523200, 1061105705492152320000, 114599416193152450560000

Offset: 0

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

Has a square root singularity.

G. C. Greubel, Table of n, a(n) for n = 0..330
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 669

Cf. A052711, A052712, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052721, A052722, A052723.

Cf. A000108, A025225.

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

Table[n!*2^n*CatalanNumber[n-1] + Boole[n==0], {n,0,30}] (* G. C. Greubel, May 29 2022 *)

[2^n*factorial(n)*catalan_number(n-1) + bool(n==0)/2 for n in (0..30)] # G. C. Greubel, May 29 2022

D-finite with recurrence: a(1)=2, a(n+1) = 4*(2*n -1)*a(n).
a(n+1) = 1/4*8^(n+1)*Gamma(n+1/2)/Pi^(1/2)
a(n+1) = ((2*n)!/n!)*2^(n+1). - Zerinvary Lajos, Sep 25 2006
a(n) = n!*A025225(n). - R. J. Mathar, Oct 18 2013
G.f.: (1- 2F0([1,-1/2], [], 8*x))/2. - R. J. Mathar, Jan 25 2020

A052717 Expansion of e.g.f. x(1 - sqrt(1 - 4x))/2.

Original entry on oeis.org

0, 0, 2, 6, 48, 600, 10080, 211680, 5322240, 155675520, 5189184000, 194075481600, 8045310873600, 366061644748800, 18134130709094400, 971471287987200000, 55956746188062720000, 3448334483839365120000

Offset: 0

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

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 673

Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052718, A052719, A052720, A052721, A052722, A052723.

Cf. A000108, A052711.

[n le 1 select 0 else Factorial(n)*Catalan(n-2): n in [0..30]]; // G. C. Greubel, May 28 2022

spec := [S,{C=Union(B,Z),B=Prod(C,C),S=Prod(Z,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
with(combinat):with(combstruct):a[0]:=0:for n from 1 to 30 do a[n]:=sum((count(Permutation(n*2-2),size=n-1)),j=0..n) od: seq(a[n], n=0..22); # Zerinvary Lajos, May 03 2007

With[{nn=20},CoefficientList[Series[x (1-Sqrt[1-4x])/2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 20 2015 *)
Table[Boole[n==1] + n!*CatalanNumber[n-2], {n, 0, 30}] (* G. C. Greubel, May 28 2022 *)

combinat::catalan(n)*(n+2)! $ n = 0..15; // Zerinvary Lajos, Feb 15 2007

[bool(n==1)/2 + factorial(n)*catalan_number(n-2) for n in (0..30)] # G. C. Greubel, May 28 2022

Recurrence: a(1)=0, a(3)=6, a(2)=2, n*a(n+1) = (4*n^2 - 2*n - 6)*a(n).
a(n) = n!*A000108(n-2) = A052711(n), n > 2. - R. J. Mathar, Oct 26 2013
G.f.: x*(d/dx)(x^2 * Hypergeometric2F0([1, 1/2], [], 4*x)). - G. C. Greubel, May 28 2022

A052719 Expansion of e.g.f. (1 - 2xsqrt(1-4x))(1 - sqrt(1-4*x))/4.

Original entry on oeis.org

0, 0, 0, 6, 72, 1080, 20160, 453600, 11975040, 363242880, 12454041600, 476367091200, 20113277184000, 929233405900800, 46630621823385600, 2525825348766720000, 146886458743664640000, 9127944221927731200000

Offset: 0

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

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 675

Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052720, A052721, A052722, A052723.

Cf. A000108, A000245.

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

Table[If[n<2, 0, 3*(n-2)*(n-1)!*CatalanNumber[n-2]], {n,0,30}] (* G. C. Greubel, May 28 2022 *)

[0,0]+[3*(n-2)*factorial(n-1)*catalan_number(n-2) for n in (2..30)] # G. C. Greubel, May 28 2022

D-finite with recurrence: a(1)=0, a(2)=0, a(3)=6, a(n+2) = (2 + 5*n)*a(n+1) + (6 + 2*n - 4*n^2)*a(n)
a(n) = n!*A000245(n-2). - R. J. Mathar, Oct 26 2013
From G. C. Greubel, May 28 2022: (Start)
G.f.: 6*x^3*Hypergeometric2F0([2, 3/2], [], 4*x).
E.g.f.: (1/4)*(1 + 2*x - 8*x^2 - (1 + 2*x)*sqrt(1-4*x)). (End)

A052720 Expansion of e.g.f.: (1 - 4x + 3x^2)(1 - 2x - sqrt(1-4x))/2 - x^2 + 2x^3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 720, 30240, 1088640, 39916800, 1556755200, 65383718400, 2964061900800, 144815595724800, 7602818775552000, 427447366714368000, 25646842002862080000, 1636734826000834560000, 110752389892723138560000

Offset: 0

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

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 676

Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052721, A052722, A052723.

Cf. A000108, A003517.

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

Table[If[n<6, 0, 6*(n-2)!*Binomial[n-4, 2]*CatalanNumber[n-3]], {n,0,30}] (* G. C. Greubel, May 28 2022 *)

def A052720(n):
    if (n<6): return 0
    else: return 6*factorial(n-2)*binomial(n-4,2)*catalan_number(n-3)
[A052720(n) for n in (0..30)] # G. C. Greubel, May 28 2022

D-finite with recurrence: a(1)=0; a(2)=0; a(4)=0; a(3)=0; a(5)=0; a(6)=720; a(n+3) = (10+8*n)*a(n+2) + (22-27*n-19*n^2)*a(n+1) - (60-66*n+6*n^2+12*n^3)*a(n).
a(n) = n!*A003517(n-4). - R. J. Mathar, Oct 18 2013
From G. C. Greubel, May 28 2022: (Start)
G.f.: 6!*x^6*Hypergeometric2F0([3, 7/2], [], 4*x).
E.g.f.: (1/2)*(1 - 6*x + 9*x^2 - 2*x^3 - (1 - 4*x + 3*x^2)*sqrt(1-4*x)). (End)

A052721 Expansion of e.g.f. x(1-2x)(1 - 2x - sqrt(1-4*x))/2 - x^3.

Original entry on oeis.org

0, 0, 0, 0, 0, 120, 2880, 70560, 1935360, 59875200, 2075673600, 79913433600, 3387499315200, 156883562035200, 7884404656128000, 427447366714368000, 24869664972472320000, 1545805113445232640000, 102232975285590589440000

Offset: 0

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

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 677

Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052722, A052723.

Cf. A000108, A002057.

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

Table[If[n<5, 0, 2*n*(n-2)!*(n-4)*CatalanNumber[n-3]], {n,0,30}] (* G. C. Greubel, May 28 2022 *)

def A052721(n):
    if (n<5): return 0
    else: return 2*n*factorial(n-2)*(n-4)*catalan_number(n-3)
[A052721(n) for n in (0..30)] # G. C. Greubel, May 28 2022

D-finite with recurrence: a(1)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=120, a(6)=2880, (n+2)*a(n+2) = (6*n^2 + 8*n - 8)*a(n+1) + (40 + 44*n = 4*n^2 - 8*n^3)*a(n).
a(n) = 2*Pi^(-1/2)*4^(n-3)*Gamma(n-5/2)*n*(n-4) for n>3.  - Mark van Hoeij, Oct 30 2011
a(n) = n!*A002057(n-5). - R. J. Mathar, Oct 18 2013
From G. C. Greubel, May 28 2022: (Start)
G.f.: 4!*x*(d/dx)( x^5 * Hypergeometric2F0([2, 5/2], [], 4*x) ).
E.g.f.: (x/2)*(1 - 4*x + 2*x^2 - (1-2*x)*sqrt(1-4*x)). (End)

A052722 Expansion of e.g.f. (1 - 2x - sqrt(1-4x))^2 * (1 - sqrt(1-4*x))/8.

Original entry on oeis.org

0, 0, 0, 0, 0, 120, 3600, 100800, 3024000, 99792000, 3632428800, 145297152000, 6351561216000, 301699157760000, 15487223431680000, 854894733428736000, 50516506975334400000, 3182539939446067200000, 212985365178313728000000

Offset: 0

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

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 678

Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052721, A052723.

Cf. A000108, A000344.

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

With[{nn=20},CoefficientList[Series[((1-2x-Sqrt[1-4x])^2 (1-Sqrt[1-4x]))/8,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 30 2021 *)
Table[If[n<5, 0, 10*(n-2)!*Binomial[n-3,2]*CatalanNumber[n-3]], {n,0,30}] (* G. C. Greubel, May 28 2022 *)

def A052722(n):
    if (n<5): return 0
    else: return 10*factorial(n-2)*binomial(n-3,2)*catalan_number(n-3)
[A052722(n) for n in (0..30)] # G. C. Greubel, May 28 2022

D-finite with recurrence: a(0) = a(1) = a(2) = a(3) = a(4) = 0, a(5)=120, a(n+3) = (9+7*n)*a(n+2) + (14 - 19*n - 13*n^2)*a(n+1) - (20 + 22*n - 2*n^2 - 4*n^3)*a(n).
a(n) = n!*A000344(n-3). - R. J. Mathar, Oct 18 2013
From G. C. Greubel, May 28 2022: (Start)
G.f.: 5!*x^5*hypergeometric2F0([5/2, 3], [], 4*x).
E.g.f.: (1/2)*(1 - 5*x + 5*x^2 - (1 - 3*x + x^2)*sqrt(1-4*x)). (End)

A151817 a(n) = 2(2n)!/n!.

Original entry on oeis.org

2, 4, 24, 240, 3360, 60480, 1330560, 34594560, 1037836800, 35286451200, 1340885145600, 56317176115200, 2590590101299200, 129529505064960000, 6994593273507840000, 405686409863454720000, 25152557411534192640000, 1660068789161256714240000, 116204815241287969996800000

Offset: 0

Roger L. Bagula, Jan 31 2009

(n+1)*a(n) is the number of random walk labelings of the comb graph of length n+1. - Sela Fried, Aug 02 2023

G. C. Greubel, Table of n, a(n) for n = 0..364
Sela Fried and Toufik Mansour, Random walk labelings of perfect trees and other graphs, arXiv:2308.00315 [math.CO], 2023.

Cf. A052718, A001813, row sums of A155951.

Table[2*(2*n)!/n!, {n, 0, 50}] (* G. C. Greubel, Feb 21 2017 *)

a(n)=2*(2*n)!/n! \\ Charles R Greathouse IV, Dec 05 2011

From Alexander R. Povolotsky, Jul 06 2009: (Start)
a(n) = 2^(2*n + 1)* Pochhammer(1/2, n).
a(n) = 2^(2*n + 1)*Gamma(n + 1/2) / Gamma(1/2) = 2^(2*n+1)*Gamma(n+1/2)/sqrt(Pi).
a(n) = 2*(2*n - 1)*a(n-1).  (End) [Updated by Peter Luschny, Aug 02 2023]
E.g.f.: 2/(1-4*x)^(1/2).- Sergei N. Gladkovskii, Dec 05 2011
G.f.: G(0), where G(k)= 1 + 1/(1 - x*(4*k+2)/(x*(4*k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 04 2013
a(n) = A052718(n+1), n>0.
a(n) = 2*A001813(n). - R. J. Mathar, Mar 12 2017

Typo in definition corrected by N. J. A. Sloane, Jul 12 2009

New name from Sergei N. Gladkovskii, Dec 05 2011

cp's OEIS Frontend

A052712 Expansion of e.g.f. (1 + 4*x - sqrt(1-8*x))/8.

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A052723 Expansion of e.g.f. (1 - x - sqrt(1-2*x+x^2-4*x^3))/(2*x).

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A052711 Expansion of e.g.f. x*(1 - 2*x - sqrt(1-4*x))/2.

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A052713 Expansion of e.g.f. (1-sqrt(1-8*x))/2.

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A052717 Expansion of e.g.f. x*(1 - sqrt(1 - 4*x))/2.

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A052719 Expansion of e.g.f. (1 - 2*x*sqrt(1-4*x))*(1 - sqrt(1-4*x))/4.

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A052720 Expansion of e.g.f.: (1 - 4*x + 3*x^2)*(1 - 2*x - sqrt(1-4*x))/2 - x^2 + 2*x^3.

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A052712 Expansion of e.g.f. (1 + 4x - sqrt(1-8x))/8.

A052723 Expansion of e.g.f. (1 - x - sqrt(1-2x+x^2-4x^3))/(2*x).

A052711 Expansion of e.g.f. x(1 - 2x - sqrt(1-4*x))/2.

A052717 Expansion of e.g.f. x(1 - sqrt(1 - 4x))/2.

A052719 Expansion of e.g.f. (1 - 2xsqrt(1-4x))(1 - sqrt(1-4*x))/4.

A052720 Expansion of e.g.f.: (1 - 4x + 3x^2)(1 - 2x - sqrt(1-4x))/2 - x^2 + 2x^3.

A052721 Expansion of e.g.f. x(1-2x)(1 - 2x - sqrt(1-4*x))/2 - x^3.

A052722 Expansion of e.g.f. (1 - 2x - sqrt(1-4x))^2 * (1 - sqrt(1-4*x))/8.

A151817 a(n) = 2(2n)!/n!.