A007272 -id:A007272 - 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

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)

A135573 Array T(n,m) of super ballot numbers read along ascending antidiagonals.

Original entry on oeis.org

1, 3, 1, 10, 2, 2, 35, 5, 3, 5, 126, 14, 6, 6, 14, 462, 42, 14, 10, 14, 42, 1716, 132, 36, 20, 20, 36, 132, 6435, 429, 99, 45, 35, 45, 99, 429, 24310, 1430, 286, 110, 70, 70, 110, 286, 1430, 92378, 4862, 858, 286, 154, 126, 154, 286, 858, 4862

Offset: 0

R. J. Mathar, Feb 23 2008

First row is A000108. 2nd row is A007054. 3rd row and 4th column are essentially A007272.
1st column is A001700. 2nd column is essentially A000108. 3rd column is A007054.
Main diagonal is A000984.

			Array with rows n >= 0 and columns m >= 0 starts:
[n\m]  0    1    2    3    4    5    6     7     8  ...
-------------------------------------------------------
[0]    1    1    2    5   14   42  132   429  1430  ...  [A000108]
[1]    3    2    3    6   14   36   99   286   858  ...  [A007054]
[2]   10    5    6   10   20   45  110   286   780  ...  [A007272]
[3]   35   14   14   20   35   70  154   364   910  ...  [A348893]
[4]  126   42   36   45   70  126  252   546  1260  ...  [A348898]
[5]  462  132   99  110  154  252  462   924  1980  ...  [A348899]
[6] 1716  429  286  286  364  546  924  1716  3432  ...
...
Seen as a triangle:
[0] 1;
[1] 3,    1;
[2] 10,   2,   2;
[3] 35,   5,   3,  5;
[4] 126,  14,  6,  6,  14;
[5] 462,  42,  14, 10, 14, 42;
[6] 1716, 132, 36, 20, 20, 36, 132;
[7] 6435, 429, 99, 45, 35, 45, 99,  429.
.
T(20, 100000) = 2.442634...*10^60129. Asymptotic formula: 2.442627..*10^60129.

G. C. Greubel, Table of n, a(n) for the first 50 antidiagonals
E. Allen and I. Gheorghiciuc, A Weighted Interpretation for the Super Catalan Numbers, J. Int. Seq. 17 (2014) # 14.10.7, Table 1.
Ira M. Gessel, Super ballot numbers, J. Symb. Comput. vol 14, iss 2-3 (1992) pp 179-194.

Cf. A000108, A007054, A000984, A348893, A348898, A348899.

Cf. A000984 (main diagonal), A001700 (column 0), A082590 (sum of antidiagonals).

T := proc(n,m) (2*n+1)!/n!*(2*m)!/m!/(m+n+1)! ; end proc:
for d from 0 to 12 do for c from 0 to d do printf("%d, ",T(d-c,c)) ; od: od:
# Alternatively, printed as rows:
A135573 := (n, m) -> (1/(2*Pi))*int(x^m*(4-x)^(n+1/2)*x^(-1/2), x=0..4):
for n from 0 to 9 do seq(A135573(n, m), m = 0..9) od; # Peter Luschny, Nov 03 2021

T[n_, m_] := (2*n+1)!/n!*(2*m)!/m!/(m+n+1)!; Table[T[n-m, m], {n, 0, 12}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jan 06 2014, after Maple *)
T[n_, m_] := 4^(m+n) Hypergeometric2F1[1/2+n, 1/2-m, 3/2+n, 1] / ((2 n + 1) Pi);
Table[T[n - m + 1, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Peter Luschny, Nov 03 2021 *)

def T(n, m): return (-1)^m*4^(n + 1 + m)*binomial(n + 1/2, n + 1 + m)/2
for n in range(7): print([T(n, m) for m in range(9)]) # Peter Luschny, Nov 04 2021

T(n, m) = (2*n + 1)!*(2*m)! / (n!*m!*(m + n + 1)!).
From Peter Luschny, Nov 03 2021: (Start)
T(n, m) = (1/(2*Pi))*Integral_{x=0..4} x^m*(4 - x)^(n + 1/2)*x^(-1/2). These are integral representations of the n-th moment of a positive function on [0, 4]. The representations are unique.
T(n, m) = 4^(m + n)*hypergeom([1/2 + n, 1/2 - m], [3/2 + n], 1)/((2*n + 1)*Pi).
For fixed n and m -> oo: T(n, m) ~ (1/(2*Pi))*4^(n + m + 1)*(Gamma(3/2 + n) / m^(3/2 + n))*(1 - (2*n + 3)^2 / (8*m)) . (End)
T(n, m) = (-1)^m*4^(n + 1 + m)*binomial(n + 1/2, n + 1 + m)/2. - Peter Luschny, Nov 04 2021
From  Peter Bala, Mar 12 2023: (Start)
T(n,m) = 2*(2*n + 1 )/(n + m + 1) * T(n-1,m) with T(0,m) = Catalan(m), where Catalan(m) = A000108(m).
T(n,m) = Sum_{k = 0..n} (-1)^k*4^(n-k)*binomial(n,k)*Catalan(m+k) (easily verified using Maple's sumrecursion command). Thus T(n,m) is an integer. (End)

A348893 a(n) = 840(2n)!/((n + 4)!*n!).

Original entry on oeis.org

35, 14, 14, 20, 35, 70, 154, 364, 910, 2380, 6460, 18088, 52003, 152950, 458850, 1400700, 4342170, 13646820, 43421700, 139704600, 454039950, 1489251036, 4925984196, 16419947320, 55124108860, 186281471320, 633357002488, 2165672331088, 7444498638115, 25717358931670, 89254363351090

Offset: 0

Karol A. Penson, Nov 02 2021

Row 3 of array A135573.

Cf. A000108, A001622, A007054, A007272, A348898, A348899.

Maple
```
seq(840*(2*n)!/((n + 4)!*n!),n=0..30)
```

a[n_] := 4^(n + 4) Hypergeometric2F1[9/2, 1/2 - n, 11/2, 1] / (9 Pi);
Table[a[n], {n, 0, 30}] (* Peter Luschny, Nov 03 2021 *)

a(n)=35*binomial(2*n,n)/binomial(n+4,4) \\ Charles R Greathouse IV, Oct 23 2023

def A348893(n): return (-4)^(4 + n)*binomial(7/2, 4 + n)/2
print([A348893(n) for n in range(31)])  # Peter Luschny, Nov 04 2021

O.g.f: (140*z^3 - 70*z^2 + 14*z - 1 + (1 - 4*z)^(7/2))/(2*z^4).
E.g.f: 64*exp(2*z)*((-z^3 - 1/2*z^2 - 1/4*z - 3/32)*BesselI(1,2*z) + BesselI(0,2*z)*z*(z^2 + 1/4*z + 3/32))/z^3.
O.g.f. g(z) satisfies z^4*g(z)^2 + (-140*z^3 + 70*z^2 - 14*z + 1)*g(z) + 4096*z^3 - 2268*z^2 + 476*z - 35 = 0;
a(n) = Integral_{x=0..4} x^n*64*(1 - x/4)^(7/2)/(Pi*sqrt(x)). This is the integral representation as the n-th moment of a positive function on [0, 4]. The representation is unique.
Remark: this sequence is not monotonically growing with n, as a(0) > a(1) = a(2) < a(3) < a(4)... .
From Peter Luschny, Nov 03 2021: (Start)
a(n) = 14*A007272(n)/(n + 4).
a(n) ~ 105*4^n*(8*n - 81)/(n^(11/2)*sqrt(Pi)).
a(n) = 4^(n + 4)*hypergeom([9/2, 1/2 - n], [11/2], 1) / (9*Pi). (End)
a(n) = (-4)^(4 + n)*binomial(7/2, 4 + n)/2. - Peter Luschny, Nov 04 2021
From Peter Bala, Mar 10 2023: (Start)
a(n) = 35*binomial(2*n, n) - 56*binomial(2*n, n + 1) + 28*binomial(2*n, n + 2) - 8*binomial(2*n, n + 3) + binomial(2*n, n + 4). Thus this sequence is integral.
7 divides a(n) except when n == 3 (mod 7).
P-recursive: (n + 4)*a(n) = 2*(2*n - 1)*a(n-1) with a(0) = 35.
D-finite: the o.g.f. A(x) satisfies the differential equation (1 - 4*x)*A'(x) + (4 - 2*x)*A(x) - 140 = 0, with A(0) = 35. (End)
From Peter Bala, Mar 11 2023: (Start)
a(n) = Sum_{k = 0..3} (-1)^k*4^(3-k)*binomial(3,k)*Catalan(n+k) = 64*Catalan(n) - 48*Catalan(n+1) + 12*Catalan(n+2) - Catalan(n+3), where Catalan(n) = A000108(n).
a(n) is odd if n = 2^k - 4, k >= 2, otherwise a(n) is even. (End)
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = 13/70 + 4*Pi/(81*sqrt(3)).
Sum_{n>=0} (-1)^(n+1)/a(n) = 72*log(phi)/(3125*sqrt(5)) - 103/43750, where phi is the golden ratio (A001622). (End)

A361030 a(n) = 20160(3n)!/(n!*(n+3)!^2).

Original entry on oeis.org

560, 210, 504, 2352, 15840, 135135, 1361360, 15519504, 194699232, 2636552100, 38003792400, 577037174400, 9155656500480, 150853746558690, 2568167588473200, 44990491457326800, 808333317429976800, 14853124707775823700, 278470827854627007600, 5316261259042879236000

Offset: 0

Peter Bala, Mar 01 2023

Row 2 of square array A361027.
The central binomial numbers A000984(n) = (2*n)!/n!^2 have the property that 60*A000984(n) is divisible by (n + 1)*(n + 2)*(n + 3) and the result 60*(2*n)!/(n!*(n+3)!) is the super ballot number A007272(n). Similarly, the de Bruijn numbers A006480(n) = (3*n)!/n!^3 have the property that 20160*A006480(n) is divisible by ((n + 1)*(n + 2)*(n + 3))^2.
Equivalently, the central binomial numbers A000984(n) = (2*n)!/n!^2 have the property that (1*3*5)*A000984(n+3) is divisible by (2*n + 1)*(2*n + 3)*(2*n + 5). The result is always an even integer. In fact, (1/2)*(1*3*5)/((2*n + 1)*(2n + 3)*(2n + 5))*A000984(n+3) = A007272(n).
Similarly, the de Bruijn numbers A006480(n) = (3*n)!/n!^3 have the property that (1*2*4*5*7*8)*A006480(n+3) is divisible by (3*n + 1)*(3*n + 2)*(3*n + 4)*(3*n + 5)*(3*n + 7)*(3*n + 8). The result is always an integer divisible by 3.

Peter Bala, A361030 is an integer sequence

Row 2 of A361027. Cf. A006480, A007272, A361028, A361029, A361031, A361038.

a := proc(n) option remember; if n = 0 then 560 else 3*(3*n-1)*(3*n-2)/(n+3)^2*a(n-1) end if; end proc:
seq(a(n), n = 0..20);

a(n) = 20160/((n+1)*(n+2)*(n+3))^2 * (3*n)!/n!^3.
a(n) = (1/3)*(1*2*4*5*7*8) * A006480(n+3)/((3*n + 1)*(3*n + 2)*(3*n + 4)*
(3*n + 5)*(3*n + 7)*(3*n + 8)), where A006480(n) = (3*n)!/n!^3.
a(n) = (1/5)*A007272(n)*A361038(n). Using this it can be shown that a(n) is always an integer.
a(n) = (1/3)*27^(n+3)*binomial(7/3, n+3)*binomial(8/3, n+3).
a(n) ~ sqrt(3)*10080*(27^n)/(Pi*n^7).
P-recursive: (n + 3)^2*a(n) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 560.
The o.g.f. A(x) satisfies the differential equation x^2*(1 - 27*x)*A''(x) + x*(7 - 54*x)*A'(x) + (9 - 6*x)*A(x) - 5040 = 0, with A(0) = 560 and A'(0) = 210.

A361038 a(n) = 1680 * (3n)!/((2n)!*(n+3)!).

Original entry on oeis.org

280, 210, 420, 1176, 3960, 15015, 61880, 271320, 1248072, 5965050, 29414700, 148874400, 770263200, 4061212722, 21765976680, 118336861720, 651555929640, 3627981880950, 20405547069180, 115815267149400, 662742214356600

Offset: 0

Peter Bala, Mar 04 2023

Compare with the super ballot numbers A007272(n) = 60*(2*n)!/(n!*(n+3)!).

Cf. A000139, A001764, A007226, A007272, A361037, A361039, A361040.

seq( 1680 * (3*n)!/((2*n)!*(n+3)!), n = 0..20);

a(n) = 280*binomial(3*n,n) - 228*binomial(3*n,n+1) + 54*binomial(3*n,n+2) - 5*binomial(3*n,n+3). Thus a(n) is an integer.
P-recursive: 2*(n + 3)*(2*n - 1) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 280.
a(n) ~ (27/4)^n * 840*sqrt(3/Pi)/n^(7/2).
The o.g.f. satisfies the differential equation
x^2*(27*x - 4)*A''(x) + 2*x*(27*x - 7)*A'(x) + (6*x + 6)*A(x) - 1680 = 0, with A(0) = 280 and A'(0) = 210.

A361039 a(n) = 55440 * (3n)!/((2n)!*(n+4)!).

Original entry on oeis.org

2310, 1386, 2310, 5544, 16335, 55055, 204204, 813960, 3432198, 15142050, 69334650, 327523680, 1588667850, 7883530578, 39904290580, 205532444040, 1075067283906, 5701114384350, 30608320603770, 166169731127400, 911270544740325

Offset: 0

Peter Bala, Mar 04 2023

Compare with the super ballot numbers A348893(n) = 840*(2*n)!/(n!*(n+4)!).

Cf. A000139, A001764, A007226, A007272, A361037, A361038, A361041.

seq(  55440 * (3*n)!/((2*n)!*(n+4)!), n = 0..20);

a(n) = 2310*binomial(3*n,n) - 2057*binomial(3*n,n+1) + 627*binomial(3*n,n+2) - 102*binomial(3*n,n+3) + 7*binomial(3*n, n+4). Thus a(n) is an integer.
P-recursive: 2*(n + 4)*(2*n - 1) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 2310.
a(n) ~ (27/4)^n * 27720*sqrt(3/Pi)/n^(9/2).
The o.g.f. satisfies the differential equation
x^2*(27*x - 4)*A''(x) + 2*x*(27*x - 9)*A'(x) + (6*x + 8)*A(x) - 18480 = 0, with A(0) = 2310 and A'(0) = 1386.

A361035 a(n) = 9979200 * (4n)!/(n!(n+3)!^3).

Original entry on oeis.org

46200, 17325, 116424, 2134440, 67953600, 3086579925, 179961581800, 12633303042360, 1023952465972800, 93080123469333000, 9292590788015304000, 1003030870975774344000, 115656146295979953692160, 14112534648127632044761125, 1808633485822731984665865000

Offset: 0

Peter Bala, Mar 01 2023

Row 2 of A361032.
The central binomial numbers A000984(n) = (2*n)!/n!^2 have the property that 60*A000984(n) is divisible by (n + 1)*(n + 2)*(n + 3) and the result (2*n)!/(n!*(n+3)!) is the super ballot number A007272(n). Similarly, the  numbers A008977(n) = (4*n)!/n!^4 appear to have the property that 9979200*A008977(n) is divisible by ((n + 1)*(n + 2)*(n + 3))^3, leading to the present sequence. Cf. A361030.
Conjecture: a(n) is odd iff n = 2^k - 3 for some k >= 2.

Cf. A007272, A008977, A361030, A361032, A361033, A361034.

seq( 9979200 * (4*n)!/(n!*(n+3)!^3 ), n = 0..20);

a(n) = 9979200 * A008977(n)/((n+1)*(n+2)*(n+3))^3.
a(n) = (15925)*A008977(n+3)/((4*n+1)*(4*n+2)*(4*n+3)*(4*n+5)*(4*n+6)*(4*n+7)*(4*n+9)*(4*n+10)*(4*n+11)).
P-recursive: a(n) = 4*(4*n-1)*(4*n-2)*(4*n-3)/(n+3)^3 * a(n-1) with a(0) = 46200.
The o.g.f. A(x) satisfies the differential equation
x^3*(1 - 256*x)*A(x)''' + x^2*(12 - 1152*x)*A(x)'' + x*(37 - 816*x)*A(x)' + (27 - 24*x)*A(x) - 1247400 = 0 with A(0) = 46200, A'(0) = 17325 and A''(0) = 232848.
a(n) ~ 2494800*sqrt(8/Pi^3) * 2^(8*n)/n^(21/2).

cp's OEIS Frontend

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

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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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A135573 Array T(n,m) of super ballot numbers read along ascending antidiagonals.

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A348893 a(n) = 840*(2*n)!/((n + 4)!*n!).

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A348893 a(n) = 840(2n)!/((n + 4)!*n!).

A361030 a(n) = 20160(3n)!/(n!*(n+3)!^2).

A361038 a(n) = 1680 * (3n)!/((2n)!*(n+3)!).

A361039 a(n) = 55440 * (3n)!/((2n)!*(n+4)!).

A361035 a(n) = 9979200 * (4n)!/(n!(n+3)!^3).