A052712 -id:A052712 - cp's OEIS frontend

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

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)

A092676 Numerators of coefficients in the series for inverf(2x/sqrt(Pi)).

Original entry on oeis.org

1, 1, 7, 127, 4369, 34807, 20036983, 2280356863, 49020204823, 65967241200001, 15773461423793767, 655889589032992201, 94020690191035873697, 655782249799531714375489, 44737200694996264619809969

Offset: 1

Eric W. Weisstein, Mar 02 2004

Differs from A002067(n) at n = 6, 9, 12, ....

Following Blair et al., we use the notation inverf() for the inverse of the error function.

			Inverf(2x/sqrt(Pi)) = x + x^3/3 + 7x^5/30 + 127x^7/630 + 4369x^9/22680 + 34807x^11/178200 + ...
The first few coefficients are 1, 1, 7/6, 127/90, 4369/2520, 34807/16200, 20036983/7484400, 2280356863/681080400, ...

G. C. Greubel, Table of n, a(n) for n = 1..230
G. Alkauskas, Algebraic and abelian solutions to the projective translation equation, arXiv preprint arXiv:1506.08028 [math.AG], 2015-2016; Aequationes Math. 90 (4) (2016), 727-763.
J. M. Blair, C. A. Edwards and J. H. Johnson, Rational Chebyshev approximations for the inverse of the error function, Math. Comp. 30 (1976), 827-830.
L. Carlitz, The inverse of the error function, Pacific J. Math., 13 (1963), 459-470.
Eric Weisstein, Mathematica program and first 50 terms of the series
Eric Weisstein's World of Mathematics, Inverse Erf
Wikipedia, Error Function

Cf. A002067, A092677, A052712. For denominators see A132467.

c:=proc(n) option remember; if n <= 0 then 1 else add( c(k)*c(n-k-1)/((k+1)*(2*k+1)), k=0..n-1 ) fi; end;

Numerator[CoefficientList[Series[InverseErf[2*x/Sqrt[Pi]], {x, 0, 50}], x]][[2 ;; ;; 2]] (* G. C. Greubel, Jan 09 2017 *)

Edited by N. J. A. Sloane, Nov 15 2007

A293471 a(n) = [x^n] (1/(1 - 2x/(1 - 2x/(1 - 4x/(1 - 4x/(1 - 6x/(1 - 6x/(1 - ...))))))))^n, a continued fraction.

Original entry on oeis.org

1, 2, 20, 248, 3472, 53152, 878144, 15577984, 296411392, 6054973952, 132994708480, 3144712222720, 80063883022336, 2192452931723264, 64427309553434624, 2025284853319303168, 67859418068644069376, 2414526405567056052224, 90909088845844899430400, 3610058425696043667030016

Offset: 0

Ilya Gutkovskiy, Oct 09 2017

nonn

Index entries for sequences related to factorial numbers

Cf. A000165, A052712, A126674, A287899, A293470.

Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-2 Floor[(k + 1)/2] x, 1, {k, 1, n}])^n, {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[Sum[(2 k)!! x^k, {k, 0, n}]^n, {x, 0, n}], {n, 0, 19}]

a(n) ~ sqrt(Pi) * 2^(n + 1/2) * n^(n + 3/2) / exp(n-1). - Vaclav Kotesovec, Sep 16 2021

cp's OEIS Frontend

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

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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.

A293471 a(n) = [x^n] (1/(1 - 2x/(1 - 2x/(1 - 4x/(1 - 4x/(1 - 6x/(1 - 6x/(1 - ...))))))))^n, a continued fraction.