A348893 -id:A348893 - cp's OEIS frontend

A007272 Super ballot numbers: 60(2n)!/(n!(n+3)!).

10, 5, 6, 10, 20, 45, 110, 286, 780, 2210, 6460, 19380, 59432, 185725, 589950, 1900950, 6203100, 20470230, 68234100, 229514700, 778354200, 2659376850, 9148256364, 31667041260, 110248217720, 385868762020, 1357193576760, 4795417304552, 17015996887120, 60619488910365

Offset: 0

N. J. A. Sloane, Simon Plouffe, Ira M. Gessel

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

Matthew House, Table of n, a(n) for n = 0..1677
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.
Ira M. Gessel, Super ballot numbers, J. Symbolic Comp., 14 (1992), 179-194
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.
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.

Row 2 of the array A135573.

Cf. A001622, A002422, A007054, A348893.

seq(10*(2*n)!/(n!)^2/binomial(n+3,n), n=0..26); # Zerinvary Lajos, Jun 28 2007

Table[60(2n)!/(n!(n+3)!), {n, 0, 30}] (* Jean-François Alcover, Jun 02 2019 *)

a(n)=if(n<0, 0, 60*(2*n)!/n!/(n+3)!) /* Michael Somos, Feb 19 2006 */

{a(n)=if(n<0, 0, n*=2; n!*polcoeff( 10*besseli(3,2*x+x*O(x^n)), n))} /* Michael Somos, Feb 19 2006 */

def A007272(n): return -(-4)^(3 + n)*binomial(5/2, 3 + n)/2
print([A007272(n) for n in range(30)])  # Peter Luschny, Nov 04 2021

G.f.: (11-32*x+9*sqrt(1-4*x))/(1-3*x+(1-x)*sqrt(1-4*x)).
E.g.f.: Sum_{n>=0} a(n)*x^(2n)/(2n)! = 60*BesselI(3, 2x)/x^3.
E.g.f.: (BesselI(0, 2*x)*(2*x+16*x^2)-BesselI(1, 2*x)*(2+6*x+16*x^2))*exp(2*x)/x^2.
Integral representation as the n-th moment of a positive function on [0, 4]: a(n) = Integral_{x=0..4} x^n*(4-x)^(5/2)/(2*Pi*x^(1/2)) dx. This representation is unique. - Karol A. Penson, Dec 04 2001
a(n) = 10*(2*n)!*[x^(2*n)](hypergeometric([],[4],x^2)). - Peter Luschny, Feb 01 2015
(n+3)*a(n) +2*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Mar 06 2018
a(n) = -(-4)^(3+n)*binomial(5/2, 3+n)/2. - Peter Luschny, Nov 04 2021
From Amiram Eldar, Mar 24 2022: (Start)
Sum_{n>=0} 1/a(n) = 4/9 + 28*Pi/(3^5*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 38/1875 - 56*log(phi)/(5^4*sqrt(5)), where phi is the golden ratio (A001622). (End)
From Peter Bala, Mar 11 2023: (Start)
a(n) = Sum_{k = 0..2} (-1)^k*4^(2-k)*binomial(n,k)*Catalan(n+k) = 16*Catalan(n) - 8*Catalan(n+1) + Catalan(n+2), where Catalan(n) = A000108(n). Thus a(n) is an integer for all n.
a(n) is odd if n = 2^k - 3, k >= 2, else a(n) is even. (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)

A361031 a(n) = (3^3)(1245781011)(3n)!/(n!*(n+4)!^2).

Original entry on oeis.org

11550, 2772, 4620, 15840, 81675, 550550, 4492488, 42325920, 446185740, 5148297000, 63985977000, 846321189120, 11802213457650, 172255143129300, 2615726247519000, 41127042052404000, 666874986879730860, 11114583114662181000, 189866473537245687000, 3316382259894423720000

Offset: 0

Peter Bala, Mar 01 2023

Row 3 of A361027.

The central binomial numbers A000984(n) = (2*n)!/n!^2 have the property that 840*A000984(n) is divisible by (n + 1)*(n + 2)*(n + 3)*(n + 4) and the result 840*(2*n)!/(n!*(n+4)!) is the super ballot number A348893(n). Similarly, the de Bruijn numbers A006480(n) = (3*n)!/n!^3 have the property that 6652800 * A006480(n) is divisible by ((n + 1)*(n + 2)*(n + 3)*(n + 4))^2.

Row 3 of A361027. Cf. A006480, A348893, A361028, A361029, A361030.

a := proc(n) option remember; if n = 0 then 11550 else
3*(3*n-1)*(3*n-2)/(n+4)^2*a(n-1) end if; end proc:
seq(a(n), n = 0..20);

a(n) = (3^3)*(1*2*4*5*7*8*10*11)/((n+1)*(n+2)*(n+3)*(n+4))^2 * (3*n)!/n!^3.
a(n) = (1/3)*(1*2*4*5*7*8*10*11) * A006480(n+4)/((3*n + 1)*(3*n + 2)*(3*n + 4)* (3*n + 5)*(3*n + 7)*(3*n + 8)*(3*n + 10)*(3*n + 11)), where A006480(n) = (3*n)!/n!^3.
a(n) = (1/3)*27^(n+4)*binomial(10/3, n+4)*binomial(11/3, n+4).
a(n) = (1/7)*A348893(n)*A361039(n). It can be shown from this that a(n) is always an integer.
a(n) ~ sqrt(3)*3326400*(27^n)/(Pi*n^9).
P-recursive: (n + 4)^2*a(n) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 11550.
The o.g.f. A(x) satisfies the differential equation x^2*(1 - 27*x)*A''(x) + x*(9 - 54*x)*A'(x) + (16 - 6*x)*A(x) - 184800 = 0, with A(0) = 11550 and A'(0) = 2772.

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.

A348899 a(n) = 3326404^nGamma(n + 1/2)/(sqrt(Pi)*Gamma(n + 7)); super ballot numbers, row 5 of A135573.

Original entry on oeis.org

462, 132, 99, 110, 154, 252, 462, 924, 1980, 4488, 10659, 26334, 67298, 177100, 478170, 1320660, 3721860, 10680120, 31150350, 92205036, 276615108, 840090328, 2580277436, 8007757560, 25090973688, 79319852304, 252832029219, 812127124158, 2627470107570, 8558045493228

Offset: 0

Peter Luschny, Nov 02 2021

nonn

Row 5 of array A135573.

Cf. A000108, A000984, A001622, A007054, A348893.

a := n -> 332640*4^n*GAMMA(n + 1/2)/(sqrt(Pi)*GAMMA(n + 7));
seq(a(n), n = 0..29);

a[n_] := 4^(n + 6) Hypergeometric2F1[13/2, 1/2 - n, 15/2, 1] / (13 Pi);
Table[a[n], {n, 0, 29}]
Array[332640*4^#*Gamma[# + 1/2]/(Sqrt[Pi]*Gamma[# + 7]) &, 30, 0] (* Michael De Vlieger, Nov 02 2021 *)

Let A[c, k](n) = c*4^n*Gamma(n + 1/2)/(sqrt(Pi)*Gamma(n + k)). Then
  A[1, 1](n) = A000984(n).
  A[3!, 3](n) = A007054(n).
  A[5!*7, 5](n) = A348893(n).
  A[7!*66, 7](n) = a(n).
  A[c, k](n) ~ -c*2^(2*n - 1)*(k^2 - k - 2*n + 1/4)/(n^(k + 1/2)*sqrt(Pi)).
O.g.f.: ((2048*x^5 - 1686*x^4 + 765*x^3 - 178*x^2 + 21*x - 1)*sqrt(1 - 4*x) - 3496*x^5 + 2934*x^4 - 1083*x^3 + 218*x^2 - 23*x + 1)/(sqrt(1 - 4*x)*(1 + sqrt(1 - 4*x))*x^5).
E.g.f.: 1024*exp(2*x)*((-x^5 - 3/4*x^4 - 41/64*x^3 - 123/256*x^2 - 9/32*x - 15/128)*BesselI(1, 2*x) + BesselI(0, 2*x)*x*(x^4 + 1/2*x^3 + 27/64*x^2 + 9/32*x + 15/128))/x^5.
a(n) = Integral_{x=0..4} x^n*(4-x)^(11/2)/(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.
a(n) = 4^(n + 6)*hypergeom([13/2, 1/2 - n], [15/2], 1) / (13*Pi).
D-finite with recurrence (n+6)*a(n) +2*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Jul 27 2022
From Peter Bala, Mar 11 2023: (Start)
a(n) = 332640*(2*n)!/(n!*(n + 6)!).
a(n) = Sum_{k = 0..5} (-1)^k*4^(5-k)*binomial(n,k)*Catalan(n+k), where Catalan(n) = A000108(n). Thus a(n) is an integer for all n.
a(n) is odd if n = 2^k - 6, k >= 3, otherwise a(n) is even. (End)
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = 101/3465 + 52*Pi/(6561*sqrt(3)).
Sum_{n>=0} (-1)^(n+1)/a(n) = 8573/54140625 + 104*log(phi)/(78125*sqrt(5)), where phi is the golden ratio (A001622). (End)

A361041 a(n) = 1680(3n)!/(n!(2n + 4)!).

Original entry on oeis.org

70, 14, 15, 28, 70, 210, 714, 2660, 10626, 44850, 197925, 906192, 4279240, 20746936, 102898110, 520543380, 2679559018, 14007652050, 74240555865, 398363958300, 2161524522150, 11847660496770, 65540249556600, 365634339159024

Offset: 0

Peter Bala, Mar 04 2023

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

Cf. A000139, A001764, A005809, A348893, A361039, A361040.

seq( 1680*(3*n)!/(n!*(2*n + 4)!), n = 0..20);

a(n) = 70*binomial(3*n,2*n) - 196*binomial(3*n,2*n+1) + 141*binomial(3*n,2*n+2) - 65*binomial(3*n,2*n+3) + 14*binomial(3*n,2*n+4). Thus a(n) is an integer.
P-recursive: 2*(n + 2)*(2*n + 3)*a(n) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 70.
a(n) ~ (27/4)^n * 105*sqrt(3/(4*Pi))/n^(9/2).
The o.g.f. A(x) satisfies the differential equation
x^2*(4 - 27*x^4)*A''(x) + 2*x*(9 - 27*x)*A'(x) + (12 - 6*x)*A(x) - 840 = 0, with A(0) = 70 and A'(0) = 14.

cp's OEIS Frontend

A007272 Super ballot numbers: 60*(2n)!/(n!*(n+3)!).

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

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A361031 a(n) = (3^3)*(1*2*4*5*7*8*10*11)*(3*n)!/(n!*(n+4)!^2).

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

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A348899 a(n) = 332640*4^n*Gamma(n + 1/2)/(sqrt(Pi)*Gamma(n + 7)); super ballot numbers, row 5 of A135573.

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A361041 a(n) = 1680*(3*n)!/(n!*(2*n + 4)!).

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A007272 Super ballot numbers: 60(2n)!/(n!(n+3)!).

A361031 a(n) = (3^3)(1245781011)(3n)!/(n!*(n+4)!^2).

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

A348899 a(n) = 3326404^nGamma(n + 1/2)/(sqrt(Pi)*Gamma(n + 7)); super ballot numbers, row 5 of A135573.

A361041 a(n) = 1680(3n)!/(n!(2n + 4)!).