A002421 -id:A002421 - cp's OEIS frontend

A007054 Super ballot numbers: 6(2n)!/(n!(n+2)!).

3, 2, 3, 6, 14, 36, 99, 286, 858, 2652, 8398, 27132, 89148, 297160, 1002915, 3421710, 11785890, 40940460, 143291610, 504932340, 1790214660, 6382504440, 22870640910, 82334307276, 297670187844, 1080432533656, 3935861372604, 14386251913656, 52749590350072

Offset: 0

N. J. A. Sloane, Mira Bernstein, Ira M. Gessel

Hankel transform is 2n+3. The Hankel transform of a(n+1) is n+2. The sequence a(n)-2*0^n has Hankel transform A110331(n). - Paul Barry, Jul 20 2008

Number of pairs of Dyck paths of total length 2*n with heights differing by at most 1 (Gessel/Xin, p. 2). - Joerg Arndt, Sep 01 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Emily Allen and Irina Gheorghiciuc, A Weighted Interpretation for the Super Catalan Numbers, J. Int. Seq. 17 (2014) # 14.10.7.
David Callan, A combinatorial interpretation for a super-Catalan recurrence, arXiv:math/0408117 [math.CO], 2004.
David Callan, A Combinatorial Interpretation for a Super-Catalan Recurrence, Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.8.
David Callan, A variant of Touchard's Catalan number identity, arXiv preprint arXiv:1204.5704 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 10 2012
Ira M. Gessel, Letter to N. J. A. Sloane, Jul. 1992
Ira M. Gessel, Super ballot numbers, J. Symbolic Comp., 14 (1992), 179-194
Ira M. Gessel, Rational Functions With Nonnegative Integer Coefficients, 50th Séminaire Lotharingien de Combinatoire, 2003.
Ira M. Gessel and Guoce Xin, A Combinatorial Interpretation of the Numbers 6(2n)!/n!(n+2)!, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.3.
Ira M. Gessel and Guoce Xin, A Combinatorial Interpretation of The Numbers 6(2n)! /n! (n+2)!, arXiv:math/0401300v2 [math.CO], 2004.
Nicholas Pippenger and Kristin Schleich, Topological characteristics of random triangulated surfaces (section 7), Random Structures Algorithms 28 (2006) 247-288; arXiv:gr-qc/0306049, 2003.
G. Schaeffer, A combinatorial interpretation of super-Catalan numbers of order two, (2001).
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 96.

Cf. A002421, A007272, A091712, A000257, A001622.

[6*Factorial(2*n)/(Factorial(n)*Factorial(n+2)): n in [0..30]]; // Vincenzo Librandi, Aug 20 2011

seq(3*(2*n)!/(n!)^2/binomial(n+2,n), n=0..22); # Zerinvary Lajos, Jun 28 2007
A007054 := n -> 6*4^n*GAMMA(1/2+n)/(sqrt(Pi)*GAMMA(3+n)):
seq(A007054(n),n=0..28); # Peter Luschny, Dec 14 2015

Table[6(2n)!/(n!(n+2)!),{n,0,30}] (* or *) CoefficientList[Series[ (-1+Sqrt[1-4*x]+(6-4*Sqrt[1-4*x])*x)/(2*x^2),{x,0,30}],x] (* Harvey P. Dale, Oct 05 2011 *)

a(n)=6*(2*n)!/(n!*(n+2)!); /* Joerg Arndt, Sep 01 2012 */

def A007054(n): return (-4)^(2 + n)*binomial(3/2, 2 + n)/2
print([A007054(n) for n in range(29)])  # Peter Luschny, Nov 04 2021

G.f.: c(x)*(4-c(x)), where c(x) = g.f. for Catalan numbers A000108; Convolution of Catalan numbers with negative Catalan numbers but -C(0)=-1 replaced by 3. - Wolfdieter Lang
E.g.f. in Maple notation: exp(2*x)*(4*x*(BesselI(0, 2*x)-BesselI(1, 2*x))-BesselI(1, 2*x))/x. Integral representation as n-th moment of a positive function on [0, 4], in Maple notation: a(n)=int(x^n*(4-x)^(3/2)/x^(1/2), x=0..4)/(2*Pi), n=0, 1, ... This representation is unique. - Karol A. Penson, Oct 10 2001
E.g.f.: Sum_{n>=0} a(n)*x^(2*n) = 3*BesselI(2, 2x).
a(n) = A000108(n)*6/(n+2). - Philippe Deléham, Oct 30 2007
a(n+1) = 2*(A000108(n+2) - A000108(n+1))/(n+1). - Paul Barry, Jul 20 2008
G.f.: ((6-4*sqrt(1-4*x))*x+sqrt(1-4*x)-1)/(2*x^2) - Harvey P. Dale, Oct 05 2011
a(n) = 4*A000108(n) - A000108(n+1) (Gessel/Xin, p. 2). - Joerg Arndt, Sep 01 2012
D-finite with recurrence (n+2)*a(n) +2*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Dec 03 2012
G.f.: 1/(x^2*G(0)) + 3/x - (1/2)/x^2, where G(k) = 1 + 1/(1 - 2*x*(2*k+3)/(2*x*(2*k+3) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013
G.f.: 3/x - 1/(2*x^2) + G(0)/(4*x^2), where G(k) = 1 + 1/(1 - 2*x*(2*k-3)/(2*x*(2*k-3) + (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 18 2013
0 = a(n)*(+16*a(n+1) - 14*a(n+2)) + a(n+1)*(+6*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Sep 18 2014
A002421(n+2) = 2*a(n) for all n in Z. - Michael Somos, Sep 18 2014
a(n) = 3*(2*n)!*[x^(2*n)]hypergeometric([],[3],x^2). - Peter Luschny, Feb 01 2015
a(n) = 6*4^n*Gamma(1/2+n)/(sqrt(Pi)*Gamma(3+n)). - Peter Luschny, Dec 14 2015
a(n) = (-4)^(2 + n)*binomial(3/2, 2 + n)/2. - Peter Luschny, Nov 04 2021
From Amiram Eldar, May 16 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 + 20*Pi/(81*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 3/25 - 8*log(phi)/(25*sqrt(5)), where phi is the golden ratio (A001622). (End)
a(n-1) = 3*A000984(n)/((2*n-1)*(n+1)). - R. J. Mathar, Jul 12 2024

Corrected and extended by Vincenzo Librandi, Aug 20 2011

A071724 a(n) = 3*binomial(2n, n-1)/(n+2), n > 0, with a(0)=1.

Original entry on oeis.org

1, 1, 3, 9, 28, 90, 297, 1001, 3432, 11934, 41990, 149226, 534888, 1931540, 7020405, 25662825, 94287120, 347993910, 1289624490, 4796857230, 17902146600, 67016296620, 251577050010, 946844533674, 3572042254128, 13505406670700

Offset: 0

N. J. A. Sloane, Jun 06 2002

Number of standard tableaux of shape (n+1,n-1) (n>=1). - Emeric Deutsch, May 30 2004
From Gus Wiseman, Apr 12 2019: (Start)
Also the number of integer partitions (of any positive integer) such that n is the maximum number of unit steps East or South in the Young diagram starting from the upper-left square and ending in a boundary square in the lower-right quadrant. Also the number of integer partitions fitting in a triangular partition of length n but not of length n - 1. For example, the a(0) = 1 through a(4) = 9 partitions are:
  ()  (1)  (2)   (3)
           (11)  (22)
           (21)  (31)
                 (32)
                 (111)
                 (211)
                 (221)
                 (311)
                 (321)
(End)
The sequence (-1)^(n+1)*a(n), for n >= 1 and +1 for n = 0, is the so-called Z-sequence of the Riordan triangle A158909. For the notion of Z- and A-sequences for Riordan arrays see the W. Lang link under A006232 with details and references. - Wolfdieter Lang, Oct 22 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Joerg Arndt, The a(4)=28 Young tableaux of shape [5, 3].
FindStat, St000384: The maximal part of the shifted composition of an integer partition.
Spencer J. Franks, Pamela E. Harris, Kimberly Harry, Jan Kretschmann, and Megan Vance, Counting Parking Sequences and Parking Assortments Through Permutations, arXiv:2301.10830 [math.CO], 2023.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 21.
Zhicong Lin, David G.L. Wang, and Tongyuan Zhao, A decomposition of ballot permutations, pattern avoidance and Gessel walks, arXiv:2103.04599 [math.CO], 2021.
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016.
Gus Wiseman, Young diagrams of all integer partitions fitting in a triangular partition of length n but not of length n - 1, n = 1...4.

Cf. A000245, A001622, A002421, A030237, A051924, A096771, A115720, A158909.
Cf. A325169, A325188, A325193, A325195, A325200.
Number of times n appears in A065770.
Column sums of A325189.
Row sums of A030237.

[1] cat [3*Binomial(2*n,n-1)/(n+2): n in [1..29]]; // Vincenzo Librandi, Jul 12 2017

A071724:= n-> 3*binomial(2*n, n-1)/(n+2); 1,seq(A071724(n), n=1..30); # G. C. Greubel, Mar 17 2021

Join[{1}, Table[3Binomial[2n, n-1]/(n+2), {n,1,30}]] (* Vincenzo Librandi, Jul 12 2017 *)
nn=7;
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
allip=Join@@Table[IntegerPartitions[n],{n,0,nn*(nn+1)/2}];
Table[Length[Select[allip,otbmax[#]==n&]],{n,0,nn}] (* Gus Wiseman, Apr 12 2019 *)

a(n)=if(n<1,n==0,3*(2*n)!/(n+2)!/(n-1)!)

[1]+[3*n*catalan_number(n)/(n+2) for n in (1..30)] # G. C. Greubel, Mar 17 2021

a(n) = A000245(n), n>0.
G.f.: (C(x)-1)*(1-x)/x = (1 + x^2 * C(x)^3)*C(x), where C(x) is g.f. for Catalan numbers, A000108.
G.f.: ((1-sqrt(1-4*x))/(2*x)-1)*(1-x)/x = A(x) satisfies x^2*A(x)^2 + (x-1)*(2*x-1)*A(x) + (x-1)^2 = 0.
G.f.: 1 + x*C(x)^3, where C(x) is g.f. for the Catalan numbers (A000108). Sequence without the first term is the 3-fold convolution of the Catalan sequence. - Emeric Deutsch, May 30 2004
a(n) is the n-th moment of the function defined on the segment (0, 4) of x axis: a(n) = Integral_{x=0..4} x^n*(-x^(1/2)*cos(3*arcsin((1/2)*x^(1/2)))/Pi) dx, n=0, 1... . - Karol A. Penson, Sep 29 2004
D-finite with recurrence -(n+2)*(n-1)*a(n) + 2*n*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Jul 10 2017
a(n) ~ c*2^(2*n)*n^(-3/2), where c = 3/sqrt(Pi). - Stefano Spezia, Sep 23 2022
From Amiram Eldar, Sep 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 14*(Pi/(3*sqrt(3)) + 1)/9.
Sum_{n>=0} (-1)^n/a(n) = 18/25 - 164*log(phi)/(75*sqrt(5)), where phi is the golden ratio (A001622). (End)

A002422 Expansion of (1-4*x)^(5/2).

Original entry on oeis.org

1, -10, 30, -20, -10, -12, -20, -40, -90, -220, -572, -1560, -4420, -12920, -38760, -118864, -371450, -1179900, -3801900, -12406200, -40940460, -136468200, -459029400, -1556708400, -5318753700, -18296512728, -63334082520

Offset: 0

N. J. A. Sloane

sign

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 55.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 164.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
N. J. A. Sloane, Notes on A984 and A2420-A2424

Cf. A007054, A004001, A002420, A002421, A002423, A002424, A007272, A001622.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-4*x)^(5/2) )); // G. C. Greubel, Jul 03 2019

A002422 := n -> -(15/8)*4^n*GAMMA(n-5/2)/(sqrt(Pi)*GAMMA(1+n)):
seq(A002422(n), n=0..26); # Peter Luschny, Dec 14 2015

CoefficientList[Series[(1-4x)^{5/2},{x,0,30}],x] (* Vincenzo Librandi, Jun 11 2012 *)

vector(30, n, n--; (-4)^n*binomial(5/2, n)) \\ G. C. Greubel, Jul 03 2019

[(-4)^n*binomial(5/2, n) for n in (0..30)] # G. C. Greubel, Jul 03 2019

a(n+3) = -2 * A007272(n).
a(n) = Sum_{m=0..n} binomial(n, m) * K_m(6), where K_m(x) = K_m(n, 2, x) is a Krawtchouk polynomial. - Alexander Barg (abarg(AT)research.bell-labs.com).
a(n) ~ -15/8*Pi^(-1/2)*n^(-7/2)*2^(2*n)*{1 + 35/8*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 22 2001
a(n) = -(15/8)*4^n*Gamma(n-5/2)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
a(n) = (-4)^n*binomial(5/2, n). - Peter Luschny, Oct 22 2018
D-finite with recurrence: n*a(n) +2*(-2*n+7)*a(n-1)=0. - R. J. Mathar, Jan 16 2020
From Amiram Eldar, Mar 24 2022: (Start)
Sum_{n>=0} 1/a(n) = 32/45 - 14*Pi/(3^5*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 2144/1875 - 28*log(phi)/(5^4*sqrt(5)), where phi is the golden ratio (A001622). (End)

A002423 Expansion of (1-4*x)^(7/2).

Original entry on oeis.org

1, -14, 70, -140, 70, 28, 28, 40, 70, 140, 308, 728, 1820, 4760, 12920, 36176, 104006, 305900, 917700, 2801400, 8684340, 27293640, 86843400, 279409200, 908079900, 2978502072, 9851968392, 32839894640

Offset: 0

N. J. A. Sloane

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 55.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 164.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Notes on A984 and A2420-A2424

Cf. A007054, A004001, A002420, A002421, A002422, A002424, A007272, A001622.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-4*x)^(7/2) )); // G. C. Greubel, Jul 03 2019

A002423 := n -> (105/16)*4^n*GAMMA(-7/2+n)/(sqrt(Pi)*GAMMA(1+n)):
seq(A002423(n), n=0..27); # Peter Luschny, Dec 14 2015

CoefficientList[Series[(1-4*x)^(7/2),{x,0,30}],x] (* Jean-François Alcover, Mar 21 2011 *)
Table[(4^(-1+x) Pochhammer[-(7/2),-1+x])/Pochhammer[1,-1+x],{x,30}] (* Harvey P. Dale, Jul 13 2011 *)

vector(30, n, n--; (-4)^n*binomial(7/2, n)) \\ G. C. Greubel, Jul 03 2019

[(-4)^n*binomial(7/2, n) for n in (0..30)] # G. C. Greubel, Jul 03 2019

a(n) = Sum_{m=0..n} binomial(n, m) * K_m(8), where K_m(x) = K_m(n, 2, x) is a Krawtchouk polynomial. - Alexander Barg (abarg(AT)research.bell-labs.com)
a(n) ~ 105*4^(n-2)/(sqrt(Pi)*n^(9/2)). - Vaclav Kotesovec, Jul 28 2013
a(n) = (105/16)*4^n*Gamma(-7/2+n)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
a(n) = (-4)^n * binomial(7/2, n). - G. C. Greubel, Jul 03 2019
D-finite with recurrence: n*a(n) +2*(-2*n+9)*a(n-1)=0. - R. J. Mathar, Jan 16 2020
From Amiram Eldar, Mar 24 2022: (Start)
Sum_{n>=0} 1/a(n) = 36/35 + 2*Pi/(3^4*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 23932/21875 - 36*log(phi)/(5^5*sqrt(5)), where phi is the golden ratio (A001622). (End)

A002424 Expansion of (1-4*x)^(9/2).

Original entry on oeis.org

1, -18, 126, -420, 630, -252, -84, -72, -90, -140, -252, -504, -1092, -2520, -6120, -15504, -40698, -110124, -305900, -869400, -2521260, -7443720, -22331160, -67964400, -209556900, -653817528, -2062039896, -6567978928, -21111360840

Offset: 0

N. J. A. Sloane

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 55.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 164.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alexander Barg, Stolarsky's invariance principle for finite metric spaces, arXiv:2005.12995 [math.CO], 2020.
N. J. A. Sloane, Notes on A984 and A2420-A2424.

Cf. A001622, A002420, A002421, A002422, A002423, A004001, A007054, A007272.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-4*x)^(9/2) )); // G. C. Greubel, Jul 03 2019

A002424 := n -> -(945/32)*4^n*GAMMA(-9/2+n)/(sqrt(Pi)*GAMMA(1+n)):
seq(A002424(n),n=0..28); # Peter Luschny, Dec 14 2015

CoefficientList[Series[(1-4x)^(9/2),{x,0,30}],x] (* Harvey P. Dale, Dec 27 2011 *)

my(x='x+O('x^30)); Vec((1-4*x)^(9/2)) \\ Altug Alkan, Dec 14 2015

vector(30, n, n--; (-4)^n*binomial(9/2, n)) \\ G. C. Greubel, Jul 03 2019

[(-4)^n*binomial(9/2, n) for n in (0..30)] # G. C. Greubel, Jul 03 2019

a(n) = Sum_{m=0..n} binomial(n, m) * K_m(10), where K_m(x) = K_m(n, 2, x) is a Krawtchouk polynomial. - Alexander Barg, abarg(AT)research.bell-labs.com.
a(n) = -(945/32)*4^n*Gamma(-9/2+n)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
a(n) = (-4)^n*binomial(9/2, n). - G. C. Greubel, Jul 03 2019
D-finite with recurrence: n*a(n) +2*(-2*n+11)*a(n-1)=0. - R. J. Mathar, Jan 16 2020
From Amiram Eldar, Mar 25 2022: (Start)
Sum_{n>=0} 1/a(n) = 32/35 - 22*Pi/(3^7*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 1050752/984375 - 44*log(phi)/(5^6*sqrt(5)), where phi is the golden ratio (A001622). (End)

A071721 Expansion of (1+x^2C^2)C^2, where C = (1-(1-4x)^(1/2))/(2x) is g.f. for Catalan numbers, A000108.

Original entry on oeis.org

1, 2, 6, 18, 56, 180, 594, 2002, 6864, 23868, 83980, 298452, 1069776, 3863080, 14040810, 51325650, 188574240, 695987820, 2579248980, 9593714460, 35804293200, 134032593240, 503154100020, 1893689067348, 7144084508256

Offset: 0

N. J. A. Sloane, Jun 06 2002

nonn

a(n) = A138156(n) - 4*A138156(n-1). - Alzhekeyev Ascar M, Jul 19 2011
Apparently, for n>=1, the sum of the heights of the first and last peaks in all Dyck n-paths (in paths with one peak the height counts as both first and last). - David Scambler, Oct 05 2012
For n>=1, a(n) is the total number of nonempty subtrees over all binary trees having n+1 internal nodes.  Here, a binary tree is a full (each node has two or zero children), rooted, plane (ordered), unlabeled tree.  An empty subtree is a tree attached to the root that consists only of an external node.  a(n) = 2*A002057(n-2) + A068875(n). - Geoffrey Critzer, Sep 16 2013
From Colin Defant, Sep 15 2018: (Start)
a(n) is the number of permutations pi of [n+1] such that s(pi) avoids the patterns 132, 231, 312, and 321, where s denotes West's stack-sorting map.
a(n) is the number of permutations on [n+1] that avoid the patterns 1342, 2341, 3142, 3241, 3412, and 3421. (End)

			G.f. = 1 + 2*x + 6*x^2 + 18*x^3 + 56*x^4 + 180*x^5 + 594*x^6 + 2002*x^7 + ... - _Michael Somos_, Apr 22 2022

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Stoyan Dimitrov, On permutation patterns with constrained gap sizes, arXiv:2002.12322 [math.CO], 2020.
Ling Gao, Graph assembly for spider and tadpole graphs, Master's Thesis, Cal. State Poly. Univ. (2023). See p. 32.
Boram Park and Seonjeong Park, Shellable posets arising from even subgraphs of a graph, arXiv preprint arXiv:1705.06423 [math.CO], 2017.
Seonjeong Park, Real toric manifolds and shellable posets arising from graphs, 2018.

Cf. A000108, A000245, A001622, A002421, A068875.
Row sums of triangles A319251, A319252.
gf=(1+x^2*C^2)*C^m: A000782 (m=1), this sequence (m=2), A071722 (m=3), A071723 (m=4).

a := n -> `if`(n=0, 1, 6*binomial(2*n, n-1)/(n+2));
seq(a(n), n=0..24); # Peter Luschny, Jun 28 2018

Join[{1},Table[6n CatalanNumber[n]/(n+2),{n,30}]] (* Harvey P. Dale, Jun 05 2012 *)
nn=20;t=(1-(1-4x)^(1/2))/(2x);CoefficientList[Series[D[1+x (y t -y+1)^2,y]/.y->1,{x,0,nn}],x] (* Geoffrey Critzer, Sep 16 2013 *)

{a(n) = if(n<1, n==0, 6*n*(2*n)!/(n!*(n + 1)!*(n + 2)))}; /* Michael Somos, Apr 22 2022 */

a = lambda n: n*(n+1)*hypergeometric([1-n, 2-n], [4], 1) if n>0 else 1
[simplify(a(n)) for n in range(25)] # Peter Luschny, Nov 19 2014

a(n) = 6n * (2n)! / [(n+2)n!(n+1)! ], n>0. In terms of Catalan numbers (A000108), a(n) = 6n*Cat(n)/(n+2), n>0. - Ralf Stephan, Mar 11 2004
a(n) = n*(n+1)*hypergeom([1-n, 2-n], [4], 1) for n>=1. - Peter Luschny, Nov 19 2014
D-finite with recurrence -(n+2)*(n-1)*a(n) +2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Jul 18 2017
a(n) = 2*Cat(n+1) - 2*Cat(n) = 2*A000245(n) for n>=1. - Colin Defant, Jun 27 2018
From Amiram Eldar, Mar 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 23/18 + 7*Pi/(27*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 43/50 - 82*sqrt(5)*log(phi)/375, where phi is the golden ratio (A001622). (End)
From Michael Somos, Apr 22 2022: (Start)
G.f.: (1 - 3*x + x^2 - (1 - x) * sqrt(1 - 4*x))/x^2.
G.f.: (2 - 2*x + x^2)/(1 - 3*x + x^2 + (1 - x)*sqrt(1 - 4*x)).
G.f.: 1 + 1/((1 - x)/(1 - sqrt(1 - 4*x)) - 1/2).
a(n) = b(n+1) - b(n) for all n in Z if b(0) = 2, b(-1) = -1, a(0) = 0, a(-1) = 3, a(-2) = -1 where b = A068875.
0 = a(n)*(+16*a(n+1) -58*a(n+2) +18*a(n+3)) +a(n+1)*(+18*a(n+1) +15*a(n+2) -13*a(n+3)) +a(n+2)*(+3*a(n+2) +a(n+3)) for all n in Z if a(0) = 0, a(-1) = 3, a(-2) = -1. (End)

A020923 Expansion of (1-4*x)^(11/2).

Original entry on oeis.org

1, -22, 198, -924, 2310, -2772, 924, 264, 198, 220, 308, 504, 924, 1848, 3960, 8976, 21318, 52668, 134596, 354200, 956340, 2641320, 7443720, 21360240, 62300700, 184410072, 553230216, 1680180656, 5160554872, 16015515120, 50181947376, 158639704608, 505664058438

Offset: 0

N. J. A. Sloane

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A001622, A002420, A002421, A002422, A002423, A002424.

A002423 := n -> (10395/64)*4^n*GAMMA(-11/2+n)/(sqrt(Pi)*GAMMA(1+n)):
seq(A002423(n),n=0..28); # Peter Luschny, Dec 14 2015

CoefficientList[Series[(1 - 4*x)^(11/2), {x,0,50}], x] (* G. C. Greubel, Feb 15 2017 *)

my(x='x+O('x^50)); Vec((1-4*x)^(11/2)) \\ G. C. Greubel, Feb 15 2017

a(n) = (10395/64)*4^n*Gamma(-11/2+n)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
D-finite with recurrence: n*a(n) +2*(-2*n+13)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
From Amiram Eldar, Mar 25 2022: (Start)
a(n) = (-4)^n*binomial(11/2, n).
Sum_{n>=0} 1/a(n) = 1124/1155 + 26*Pi/(3^8*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 56972276/54140625 - 52*log(phi)/(5^7*sqrt(5)), where phi is the golden ratio (A001622). (End)

A020925 Expansion of (1-4*x)^(13/2).

Original entry on oeis.org

1, -26, 286, -1716, 6006, -12012, 12012, -3432, -858, -572, -572, -728, -1092, -1848, -3432, -6864, -14586, -32604, -76076, -184184, -460460, -1184040, -3121560, -8414640, -23140260, -64792728, -184410072, -532740208, -1560167752, -4626704368, -13880113104

Offset: 0

N. J. A. Sloane

sign

Robert Israel, Table of n, a(n) for n = 0..1694

Cf. A001622, A002420, A002421, A002422, A002423, A002424, A020923.

f := k -> -135135*(2*k)!/((2*k-1)*(2*k-3)*(2*k-5)*(2*k-7)*(2*k-9)*(2*k-11)*(-13+2*k)*(k!)^2):
map(f, [$0..30]); # Robert Israel, Jul 02 2018

CoefficientList[Series[(1-4*x)^(13/2), {x, 0, 50}], x] (* Amiram Eldar, Mar 25 2022 *)

my(x = 'x + O('x^40)); Vec((1-4*x)^(13/2)) \\ Michel Marcus, Jul 02 2018

a(n) = (-2)^n * Product_{i=0..n-1} (13-2*i) / n! for n>0. - R. J. Mathar, Feb 19 2008
D-finite with recurrence: n*a(n) - 2*(2*n-13)*a(n-1) = 0 for n>0. - Bruno Berselli, Jul 02 2018
a(n) ~ -135135 * 2^(2*n - 7) / (sqrt(Pi) * n^(15/2)). - Vaclav Kotesovec, Jul 02 2018
From Amiram Eldar, Mar 25 2022: (Start)
a(n) = (-4)^n*binomial(13/2, n).
Sum_{n>=0} 1/a(n) = 960/1001 - 10*Pi/(3^8*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 244659776/234609375 - 12*log(phi)/(5^7*sqrt(5)), where phi is the golden ratio (A001622). (End)

A382536 Expansion of 1/(1 - x(1 + 4x)^(3/2)).

Original entry on oeis.org

1, 1, 7, 19, 63, 221, 679, 2365, 7499, 25351, 82043, 274031, 892263, 2972127, 9686899, 32261819, 105124711, 350277365, 1140610399, 3803874525, 12372800403, 41319077557, 134176480535, 448958154449, 1454582791283, 4879992151217, 15762304059447, 53067612190093

Offset: 0

Seiichi Manyama, Mar 31 2025

a(62) is negative.

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A362154, A382537, A382538.

Cf. A002421, A382514.

R := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x*(1 + 4*x)^(3/2)); seq := [ Coefficient(f, n) : n in [0..30] ]; seq; // Vincenzo Librandi, Apr 01 2025

Table[Sum[4^(n-k)*Binomial[3*k/2,n-k],{k,0,n}],{n,0,35}] (* Vincenzo Librandi, Apr 01 2025 *)

a(n) = sum(k=0, n, 4^(n-k)*binomial(3*k/2, n-k));

a(n) = Sum_{k=0..n} 4^(n-k) * binomial(3*k/2,n-k).
D-finite with recurrence (-n+1)*a(n) +2*(-2*n+7)*a(n-1) +(n-1)*a(n-2) +2*(8*n-13)*a(n-3) +24*(4*n-9)*a(n-4) +32*(8*n-23)*a(n-5) +128*(2*n-7)*a(n-6)=0. - R. J. Mathar, Apr 02 2025
a(n) ~ 3 * (-1)^(n+1) * 2^(2*n-4) / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Apr 13 2025

A020927 Expansion of (1-4*x)^(15/2).

Original entry on oeis.org

1, -30, 390, -2860, 12870, -36036, 60060, -51480, 12870, 2860, 1716, 1560, 1820, 2520, 3960, 6864, 12870, 25740, 54340, 120120, 276276, 657800, 1614600, 4071600, 10518300, 27768312, 74760840, 204900080, 570793080, 1613966640, 4626704368, 13432367520

Offset: 0

N. J. A. Sloane

sign

Cf. A001622, A002420, A002421, A002422, A002423, A002424, A020923, A020925, A020929.

CoefficientList[Series[(1-4x)^(15/2),{x,0,30}],x] (* Harvey P. Dale, Oct 03 2012 *)

D-finite with recurrence: n*a(n) +2*(-2*n+17)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
From Amiram Eldar, Mar 25 2022: (Start)
a(n) = (-4)^n*binomial(15/2, n).
Sum_{n>=0} 1/a(n) = 972/1001 + 34*Pi/(3^10*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 18235778692/17595703125 - 68*log(phi)/(5^9*sqrt(5)), where phi is the golden ratio (A001622). (End)

cp's OEIS Frontend

A007054 Super ballot numbers: 6(2n)!/(n!(n+2)!).

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A071724 a(n) = 3*binomial(2n, n-1)/(n+2), n > 0, with a(0)=1.

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A002422 Expansion of (1-4*x)^(5/2).

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A002423 Expansion of (1-4*x)^(7/2).

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A002424 Expansion of (1-4*x)^(9/2).

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A071721 Expansion of (1+x^2*C^2)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

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A071721 Expansion of (1+x^2C^2)C^2, where C = (1-(1-4x)^(1/2))/(2x) is g.f. for Catalan numbers, A000108.

A382536 Expansion of 1/(1 - x(1 + 4x)^(3/2)).