A002423 -id:A002423 - cp's OEIS frontend

A002421 Expansion of (1-4*x)^(3/2) in powers of x.

1, -6, 6, 4, 6, 12, 28, 72, 198, 572, 1716, 5304, 16796, 54264, 178296, 594320, 2005830, 6843420, 23571780, 81880920, 286583220, 1009864680, 3580429320, 12765008880, 45741281820, 164668614552, 595340375688, 2160865067312, 7871722745208, 28772503827312

Offset: 0

N. J. A. Sloane

Terms that are not divisible by 12 have indices in A019469. - Ralf Stephan, Aug 26 2004
From Ralf Steiner, Apr 06 2017: (Start)
By analytic continuation to the entire complex plane there exist regularized values for divergent sums such as:
Sum_{k>=0} a(k)^2/8^k = 2F1(-3/2,-3/2,1,2).
Sum_{k>=0} a(k) / 2^k = -i. (End)

			G.f. = 1 - 6*x + 6*x^2 + 4*x^3 + 6*x^4 + 12*x^5 + 28*x^6 + 72*x^7 + 198*x^8 + 572*x^9 + ...

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, A002422, A002423, A002424.

Cf. A000257, A001622, A071721, A071724, A085687.

Concatenation([1], List([1..40], n-> 12*Factorial(2*n-4) /( Factorial(n)*Factorial(n-2)) )) # G. C. Greubel, Jul 03 2019

[1,-6] cat [12*Catalan(n-2)/n: n in [2..30]]; // Vincenzo Librandi, Jun 11 2012

A002421 := n -> 3*4^(n-1)*GAMMA(-3/2+n)/(sqrt(Pi)*GAMMA(1+n)):
seq(A002421(n), n=0..29); # Peter Luschny, Dec 14 2015

CoefficientList[Series[(1-4x)^(3/2),{x,0,40}],x] (* Vincenzo Librandi, Jun 11 2012 *)
a[n_]:= Binomial[ 3/2, n] (-4)^n; (* Michael Somos, Dec 04 2013 *)
a[n_]:= SeriesCoefficient[(1-4x)^(3/2), {x, 0, n}]; (* Michael Somos, Dec 04 2013 *)

{a(n) = binomial( 3/2, n) * (-4)^n}; /* Michael Somos, Dec 04 2013 */

{a(n) = if( n<0, 0, polcoeff( (1 - 4*x + x * O(x^n))^(3/2), n))}; /* Michael Somos, Dec 04 2013 */

((1-4*x)^(3/2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jul 03 2019

a(n) = Sum_{m=0..n} binomial(n, m)*K_m(4), where K_m(x) = K_m(n, 2, x) is a Krawtchouk polynomial. - Alexander Barg (abarg(AT)research.bell-labs.com)
a(n) ~ (3/4)*Pi^(-1/2)*n^(-5/2)*2^(2*n)*(1 + 15/8*n^-1 + ...). - Joe Keane (jgk(AT)jgk.org), Nov 22 2001
From Ralf Stephan, Mar 11 2004: (Start)
a(n) = 12*(2*n-4)! /(n!*(n-2)!), n > 1.
a(n) = 12*Cat(n-2)/n = 2(Cat(n-1) - 4*Cat(n-2)), in terms of Catalan numbers (A000108).
Terms that are not divisible by 12 have indices in A019469. (End)
Let rho(x)=(1/Pi)*(x*(4-x))^(3/2), then for n >= 4, a(n) = Integral_{x=0..4} (x^(n-4) *rho(x)) dx. - Groux Roland, Mar 16 2011
G.f.: (1-4*x)^(3/2) = 1 - 6*x + 12*x^2/(G(0) + 2*x); G(k) = (4*x+1)*k-2*x+2-2*x*(k+2)*(2*k+1)/G(k+1); for -1/4 <= x < 1/4, otherwise G(0) = 2*x; (continued fraction). - Sergei N. Gladkovskii, Dec 05 2011
G.f.: 1/G(0) where G(k) = 1 + 4*x*(2*k+1)/(1 - 1/(1 + (2*k+2)/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 18 2012
G.f.: G(0)/2, where G(k) = 2 + 2*x*(2*k-3)*G(k+1)/(k+1). - Sergei N. Gladkovskii, Jun 06 2013 [Edited by Michael Somos, Dec 04 2013]
0 = a(n+2) * (a(n+1) - 14*a(n)) + a(n+1) * (6*a(n+1) + 16*a(n)) for all n in Z. - Michael Somos, Dec 04 2013
A232546(n) = 3^n * a(n). - Michael Somos, Dec 04 2013
G.f.: hypergeometric1F0(-3/2;;4*x). - R. J. Mathar, Aug 09 2015
a(n) = 3*4^(n-1)*Gamma(-3/2+n)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
From Ralf Steiner, Apr 06 2017: (Start)
Sum_{k>=0} a(k)/4^k = 0.
Sum_{k>=0} a(k)^2/16^k = 32/(3*Pi).
Sum_{k>=0} a(k)^2*(k/8)/16^k = 1/Pi.
Sum_{k>=0} a(k)^2*(-k/24+1/8)/16^k = 1/Pi.
Sum_{k>=0} a(k-1)^2*(k-1/4)/16^k = 1/Pi.
Sum_{k>=0} a(k-1)^2*(2k-2)/16^k = 1/Pi.(End)
D-finite with recurrence: n*a(n) +2*(-2*n+5)*a(n-1)=0. - R. J. Mathar, Feb 20 2020
From Amiram Eldar, Mar 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 4/3 + 10*Pi/(81*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 92/75 - 4*sqrt(5)*log(phi)/125, 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)

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)

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)

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)

A020929 Expansion of (1-4*x)^(17/2).

Original entry on oeis.org

1, -34, 510, -4420, 24310, -87516, 204204, -291720, 218790, -48620, -9724, -5304, -4420, -4760, -6120, -8976, -14586, -25740, -48620, -97240, -204204, -447304, -1016600, -2386800, -5768100, -14304888, -36312408, -94143280, -248807240, -669205680, -1829162192

Offset: 0

N. J. A. Sloane

sign

Cf. A001622, A002420, A002421, A002422, A002423, A002424, A020923, A020925, A020927.

CoefficientList[Series[(1 - 4 x)^(17/2), {x, 0, 33}], x] (* Vincenzo Librandi, Jan 18 2020 *)

D-finite with recurrence: n*a(n) +2*(-2*n+19)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
From Amiram Eldar, Mar 25 2022: (Start)
a(n) = (-4)^n*binomial(17/2, n).
Sum_{n>=0} 1/a(n) = 49600/51051 - 38*Pi/(3^11*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 1542987607648/1495634765625 - 76*log(phi)/(5^10*sqrt(5)), where phi is the golden ratio (A001622). (End)

A382538 Expansion of 1/(1 - x(1 + 4x)^(7/2)).

Original entry on oeis.org

1, 1, 15, 99, 519, 3165, 19503, 115053, 688803, 4141863, 24778355, 148376447, 889216143, 5326274463, 31903872267, 191123789739, 1144894457103, 6858232252437, 41083285178247, 246102886383661, 1474237118571467, 8831178384769525, 52901735792001759

Offset: 0

Seiichi Manyama, Mar 31 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..600

Cf. A362154, A382536, A382537.

Cf. A002423.

R:=PowerSeriesRing(Rationals(), 28); Coefficients(R!( 1/(1 - x*(1 + 4*x)^(7/2)))); // Vincenzo Librandi, May 16 2025

Table[Sum[4^(n-k)* Binomial[7*k/2, n-k],{k,0,n}],{n,0,28}] (* Vincenzo Librandi, May 16 2025 *)

a(n) = sum(k=0, n, 4^(n-k)*binomial(7*k/2, n-k));

a(n) = Sum_{k=0..n} 4^(n-k) * binomial(7*k/2,n-k).

D-finite with recurrence (-n+1)*a(n) +2*(-2*n+11)*a(n-1) +(n-1)*a(n-2) +2*(16*n-25)*a(n-3) +56*(8*n-17)*a(n-4) +224*(16*n-43)*a(n-5) +4480*(4*n-13)*a(n-6) +3584*(16*n-61)*a(n-7) +14336*(8*n-35)*a(n-8) +8192*(16*n-79)*a(n-9) +32768*(2*n-11)*a(n-10)=0. - R. J. Mathar, Apr 02 2025

A020931 Expansion of (1-4*x)^(19/2).

Original entry on oeis.org

1, -38, 646, -6460, 41990, -184756, 554268, -1108536, 1385670, -923780, 184756, 33592, 16796, 12920, 12920, 15504, 21318, 32604, 54340, 97240, 184756, 369512, 772616, 1679600, 3779100, 8767512, 20907144, 51106352, 127765880, 326023280, 847660528, 2242198816

Offset: 0

N. J. A. Sloane

sign

Cf. A001622, A002420, A002421, A002422, A002423, A002424, A020923, A020925, A020927, A020929.

CoefficientList[Series[(1-4x)^(19/2),{x,0,30}],x] (* Harvey P. Dale, Jul 03 2013 *)

D-finite with recurrence: n*a(n) +2*(-2*n+21)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
From Amiram Eldar, Mar 25 2022: (Start)
a(n) = (-4)^n*binomial(19/2, n).
Sum_{n>=0} 1/a(n) = 45052/46189 + 14*Pi/(3^11*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 6955761045148/6765966796875 - 84*log(phi)/(5^11*sqrt(5)), where phi is the golden ratio (A001622). (End)

A182411 Triangle T(n,k) = (2k)!(2n)!/(k!n!*(k+n)!) with k=0..n, read by rows.

Original entry on oeis.org

1, 2, 2, 6, 4, 6, 20, 10, 12, 20, 70, 28, 28, 40, 70, 252, 84, 72, 90, 140, 252, 924, 264, 198, 220, 308, 504, 924, 3432, 858, 572, 572, 728, 1092, 1848, 3432, 12870, 2860, 1716, 1560, 1820, 2520, 3960, 6864, 12870, 48620, 9724, 5304, 4420, 4760, 6120, 8976

Offset: 0

Bruno Berselli, Apr 27 2012

This is a companion to the triangle A068555.
Row sum is 2*A132310(n-1) + A000984(n) for n>0, where A000984(n) = T(n,0) = T(n,n). Also:
T(n,1)  = -A002420(n+1).
T(n,2)  =  A002421(n+2).
T(n,3)  = -A002422(n+3) = 2*A007272(n).
T(n,4)  =  A002423(n+4).
T(n,5)  = -A002424(n+5).
T(n,6)  =  A020923(n+6).
T(n,7)  = -A020925(n+7).
T(n,8)  =  A020927(n+8).
T(n,9)  = -A020929(n+9).
T(n,10) =  A020931(n+10).
T(n,11) = -A020933(n+11).

			Triangle begins:
      1;
      2,    2;
      6,    4,    6;
     20,   10,   12,   20;
     70,   28,   28,   40,   70;
    252,   84,   72,   90,  140,  252;
    924,  264,  198,  220,  308,  504,  924;
   3432,  858,  572,  572,  728, 1092, 1848,  3432;
  12870, 2860, 1716, 1560, 1820, 2520, 3960,  6864, 12870;
  48620, 9724, 5304, 4420, 4760, 6120, 8976, 14586, 25740, 48620;
  ...
Sum_{k=0..8} T(8,k) = 12870 + 2860 + 1716 + 1560 + 1820 + 2520 + 3960 + 6864 + 12870 = 2*A132310(7) + A000984(8) = 2*17085 + 12870 = 47040.

Umberto Scarpis, Sui numeri primi e sui problemi dell'analisi indeterminata in Questioni riguardanti le matematiche elementari, Nicola Zanichelli Editore (1924-1927, third edition), page 11.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 103.

Alexander Borisov, Quotient singularities, integer ratios of factorials and the Riemann Hypothesis, arXiv:math/0505167 [math.NT], 2005; International Mathematics Research Notices, Vol. 2008, Article ID rnn052, page 2 (Theorem 2).
Ira Gessel, Integer quotients of factorials and algebraic multivariable hypergeometric series, MIT Combinatorics Seminar, September 2011 (slides).
Hans-Christian Herbig and Mateus de Jesus Gonçalves, On the numerology of trigonometric polynomials, arXiv:2311.13604 [math.HO], 2023.
Kevin Limanta and Norman Wildberger, Super Catalan Numbers, Chromogeometry, and Fourier Summation over Finite Fields, arXiv:2108.10191 [math.CO], 2021. See Table 1 p. 2 where terms are shown as an array.

Cf. A000984, A002420-A020933, A068555, A132310.

[Factorial(2*k)*Factorial(2*n)/(Factorial(k)*Factorial(n)*Factorial(k+n)): k in [0..n], n in [0..9]];

Flatten[Table[Table[(2 k)! ((2 n)!/(k! n! (k + n)!)), {k, 0, n}], {n, 0, 9}]]

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A002421 Expansion of (1-4*x)^(3/2) in powers of x.

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

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

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

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

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

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A020929 Expansion of (1-4*x)^(17/2).

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A382538 Expansion of 1/(1 - x(1 + 4x)^(7/2)).

A182411 Triangle T(n,k) = (2k)!(2n)!/(k!n!*(k+n)!) with k=0..n, read by rows.