A184423 -id:A184423 - cp's OEIS frontend

A001421 a(n) = (6n)!/((n!)^3(3*n)!).

1, 120, 83160, 81681600, 93699005400, 117386113965120, 155667030019300800, 214804163196079142400, 305240072216678400087000, 443655767845074392936328000, 656486312795713480715743268160, 985646873056680684690542988249600, 1497786250388951255453847206769124800

Offset: 0

N. J. A. Sloane, Glenn K Painter (KUPK78A(AT)prodigy.com)

nonn

Self-convolution of A092870, where A092870(n) = (12^n/n!^2) * Product_{k=0..n-1} (12k+1)*(12k+5). - Paul D. Hanna, Jan 25 2011

			G.f.: A(x) = 1 + 120*x + 83160*x^2 + 81681600*x^3 + ...
A(x)^(1/2) = 1 + 60*x + 39780*x^2 + 38454000*x^3 + ... + A092870(n)*x^n + ...

Seiichi Manyama, Table of n, a(n) for n = 0..310 (terms 0..75 from Vincenzo Librandi)
Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998 (See Eq. 31.)
R. S. Maier, Nonlinear differential equations satisfied by certain classical modular forms, arXiv:0807.1081 [math.NT], 2008_2010, p. 34 equation (7.29b).
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Cf. A092870; variants: A184423, A008977, A184892, A184896, A184898. - Paul D. Hanna, Jan 25 2011
Cf. A289292.
Cf. A145492, A145493, A145494, A008979.

[Factorial(6*n)/(Factorial(n)^3*Factorial(3*n)): n in [0..15]]; // Vincenzo Librandi, Oct 26 2011

Maple
```
f := n->(6*n)!/( (n!)^3*(3*n)!);
```

Factorial[6 n]/(Factorial[3n] Factorial[n]^3) (* Jacob Lewis (jacobml(AT)uw.edu), Jul 28 2009 *)
a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/6, 1/2, 5/6}, {1, 1}, 1728 x], {x, 0, n}] (* Michael Somos, Jul 11 2011 *)

{a(n)=(2*n)!/n!^2*(12^n/n!^2)*prod(k=0, n-1, (6*k+1)*(6*k+5))} \\ Paul D. Hanna, Jan 25 2011

O.g.f.: Hypergeometric2F1(5/12, 1/12; 1; 1728x)^2. - Jacob Lewis (jacobml(AT)uw.edu), Jul 28 2009
a(n) = binomial(2n,n) * (12^n/n!^2) * Product_{k=0..n-1} (6k+1)*(6k+5). - Paul D. Hanna, Jan 25 2011
G.f.: F(1/6, 1/2, 5/6; 1, 1; 1728*x), a hypergeometric series. - Michael Somos, Feb 28 2011
0 = y^3*z^3 - 360*y^4*z^2 + 43200*y^5*z - 1728000*y^6 - 16632*x*y^2*z^3 + 7691328*x*y^3*z^2 - 1738520064*x*y^4*z + 176027074560*x*y^5 + 92207808*x^2*y*z^3 - 69176553984*x^2*y^2*z^2 + 23624298528768*x^2*y^3*z - 2853152143441920*x^2*y^4 - 170400029184*x^3*z^3 + 224945232150528*x^3*y*z^2 - 92759146352345088*x^3*y^2*z + 11686511179538104320*x^3*y^3 where x = a(n), y = a(n+1), z = a(n+2) for all n in z. - Michael Somos, Sep 21 2014
a(n) ~ 2^(6*n - 1) * 3^(3*n) / (Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Apr 07 2018
From Peter Bala, Feb 14 2020: (Start)
a(n) = binomial(6*n,n)*binomial(5*n,n)*binomial(4*n,n) = ( [x^n](1 + x)^(6*n) ) * ( [x^n](1 + x)^(5*n) ) * ( [x^n](1 + x)^(4*n) ) = [x^n](F(x)^(120*n)), where F(x) = 1 + x + 227*x^2 + 123980*x^3 + 92940839*x^4 + 82527556542*x^5 + 81459995686401*x^6 + ...
appears to have integer coefficients. For similar results see A008979.
a(m*p^k) == a(m*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers m and k - apply Mestrovic, equation 39, p. 12.
a(n) = [(x*y*z)^n] (1 + x + y + z)^(6*n). (End)
a(n) = (8^n/n!^3)*Product_{k = 0..3*n-1} (2*k + 1). - Peter Bala, Feb 26 2023
a(n) = 24*(6*n - 1)*(2*n - 1)*(6*n - 5)*a(n-1)/n^3. - Neven Sajko, Jul 19 2023
From Karol A. Penson, Jan 21 2025: (Start)
a(n) = Integral_{x=0..1728} x^n*W(x), with W(x) =  W1(x) + W2(x) + W3(x), where
  W1(x) = hypergeometric([1/6, 1/6, 1/6], [1/3, 2/3], x/1728)/(6*sqrt(Pi)*x^(5/6)*Gamma(5/6)^3),
  W2(x) = - hypergeometric([1/2, 1/2, 1/2], [2/3, 4/3], x/1728)/(24*Pi^2*sqrt(x)), and
  W3(x) = hypergeometric([5/6, 5/6, 5/6], [4/3, 5/3], x/1728)*Gamma(5/6)^3/(1536*Pi^(7/2)*x^(1/6)). This integral representation is unique as W(x) is the solution of the Hausdorff power moment problem on x = (0, 1728). Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0, with singularity x^(-1/6), and for x > 0 is monotonically decreasing to zero at x = 1728. (End)

A184424 a(n) = (3^n/n!^2) * Product_{k=1..n} (6k-4)*(6k-5).

Original entry on oeis.org

1, 6, 252, 15288, 1089270, 84963060, 7023612960, 604604070720, 53620823521980, 4865593245513000, 449580815885401200, 42156561463105471200, 4001360292206427641400, 383704407665664889683600

Offset: 0

Paul D. Hanna, Jan 13 2011

nonn

			G.f.: A(x) = 1 + 6*x + 252*x^2 + 15288*x^3 + 1089270*x^4 +...
G.f. of A184423 equals A(x)^2:
A(x)^2 = 1 + 12*x + 540*x^2 + 33600*x^3 + 2425500*x^4 + 190702512*x^5 +...+ [(2n)!*(3n)!/n!^5]*x^n +...

G. C. Greubel, Table of n, a(n) for n = 0..492

Cf. A184423.

a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/6, 1/3, 1, 108 x], {x, 0, n}]; (* Michael Somos, Sep 26 2011 *)
Table[3^n/(n!)^2 Product[(6k-4)(6k-5),{k,n}],{n,0,20}] (* Harvey P. Dale, May 10 2019 *)

{a(n)=3^n*prod(k=1,n,(6*k-4)*(6*k-5))/n!^2}

{a(n)=polcoeff(sum(m=0,n,(2*m)!*(3*m)!/m!^5*x^m+x*O(x^n))^(1/2),n)}

Self-convolution equals A184423, where A184423(n) = (2n)!*(3n)!/n!^5:
Sum_{k=0..n} a(n-k)*a(k) = (2n)!*(3n)!/n!^5.
a(n) ~ 2^(2*n + 1/3) * 3^(3*n - 1/2) * sqrt(Pi) / (Gamma(1/3)^3 * n^(3/2)). - Vaclav Kotesovec, Jun 09 2019

A184896 a(n) = C(2n,n) * (7^n/n!^2) * Product_{k=0..n-1} (7k+1)*(7k+6).

Original entry on oeis.org

1, 84, 45864, 35672000, 32445913500, 32247604076688, 33935228690034672, 37165308416775931392, 41919854708375196052500, 48365506771435816732770000, 56812832722107710740048677120, 67715433011522917282547695380480

Offset: 0

Paul D. Hanna, Jan 25 2011

nonn

			G.f.: A(x) = 1 + 84*x + 45864*x^2 + 35672000*x^3 +...
A(x)^(1/2) = 1 + 42*x + 22050*x^2 + 16909900*x^3 +...+ A184895(n)*x^n +...

Cf. A184895; variants: A184423, A008977, A184892, A001421, A184898.

{a(n)=(2*n)!/n!^2*(7^n/n!^2)*prod(k=0,n-1,(7*k+1)*(7*k+6))}

Self-convolution of A184895, where A184895(n) = (7^n/n!^2) * Product_{k=0..n-1} (14k+1)*(14k+6).

a(n) ~ sin(Pi/7) * 2^(2*n) * 7^(3*n) / (Pi*n)^(3/2). - Vaclav Kotesovec, Oct 23 2020

A184892 a(n) = C(2n,n) * (5^n/n!^2) * Product_{k=0..n-1} (5k+1)*(5k+4).

Original entry on oeis.org

1, 40, 8100, 2310000, 768075000, 278719056000, 107022956040000, 42753018765600000, 17585519046944062500, 7397979398239787500000, 3168258657090171394750000, 1376657183877933677265000000

Offset: 0

Paul D. Hanna, Jan 25 2011

nonn

			G.f.: A(x) = 1 + 40*x + 8100*x^2 + 2310000*x^3 +...
A(x)^(1/2) = 1 + 20*x + 3850*x^2 + 1078000*x^3 +...+ A184891(n)*x^n +...

Cf. A184891, A184890, A184423, A008977, A001421, A184896, A184898.

Table[Binomial[2*n, n] * 5^n / n!^2 * Product[(5*k + 1)*(5*k + 4), {k, 0, n - 1}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 07 2020 *)

{a(n)=(2*n)!/n!^2*(5^n/n!^2)*prod(k=0,n-1,(5*k+1)*(5*k+4))}

Self-convolution of A184891, where
. A184891(n) = (5^n/n!^2) * Product_{k=0..n-1} (10k+1)*(10k+4).
a(n) ~ sqrt(5 - sqrt(5)) * 2^(2*n - 3/2) * 5^(3*n) / (Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Oct 07 2020

A184898 a(n) = C(2n,n) * (8^n/n!^2) * Product_{k=0..n-1} (8k+1)*(8k+7).

Original entry on oeis.org

1, 112, 90720, 105100800, 142542960000, 211337613527040, 331831362513530880, 542307255307827609600, 912855634598629193472000, 1571864775032876891607040000, 2755743023914838714304931102720

Offset: 0

Paul D. Hanna, Jan 25 2011

nonn

			G.f.: A(x) = 1 + 112*x + 90720*x^2 + 105100800*x^3 +...
A(x)^(1/2) = 1 + 56*x + 43792*x^2 + 50098048*x^3 +...+ A184897(n)*x^n +...

Cf. A184897; variants: A184423, A008977, A184892, A001421, A184896.

{a(n)=(2*n)!/n!^2*(8^n/n!^2)*prod(k=0,n-1,(8*k+1)*(8*k+7))}

Self-convolution of A184897, where A184897(n) = (8^n/n!^2) * Product_{k=0..n-1} (16k+1)*(16k+7).

a(n) ~ sqrt(2-sqrt(2)) * 2^(11*n - 1) / (Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Oct 05 2020

A284756 (n + 1)^3a(n + 1) = 2(2n + 1)(5n^2 + 5n + 2)a(n) - 8n(7n^2 + 1)a(n - 1) + 22n(n - 1)(2n - 1)a(n - 2), with a(0) = 1, a(1) = 4 and a(2) = 28.

Original entry on oeis.org

1, 4, 28, 268, 3004, 36784, 476476, 6418192, 88986172, 1261473136, 18200713168, 266393373712, 3945664966204, 59029237351504, 890697897694192, 13539585443232688, 207149418061499452, 3187355160332835184, 49290960047575223824, 765703166164798253392

Offset: 0

Seiichi Manyama, Apr 02 2017

nonn

This sequence is c_11 in the 2015 paper of Cooper et al.

Seiichi Manyama, Table of n, a(n) for n = 0..819
S. Cooper, Ramanujan's theories of elliptic functions to alternative bases, and beyond, talk slides, Askey 80 Conference, 2013.
S. Cooper, J. Ge and D. Ye, Hypergeometric transformation formulas of degrees 3, 7, 11 and 23, Journal of Mathematical Analysis and Applications, Volume 421, Issue 2, 15 January 2015, Pages 1358-1376.

Cf. A184423 (c_3), A183204 (c_7), this sequence (c_11).

RecurrenceTable[{(n+1)^3*a[n+1] == 2*(2*n+1)*(5*n^2+5*n+2)*a[n] - 8*n*(7*n^2+1)*a[n-1] + 22*n*(n-1)*(2*n-1)*a[n-2], a[0]==1, a[1]==4, a[2]==28}, a, {n, 0, 20}] (* Vaclav Kotesovec, Apr 02 2017 *)

def A284756(n)
  a, b, c, i = 0, 0, 1, -1
  ary = [0, 0]
  while i < n
    i += 1
    j = 2 * (2 * i + 1) * (5 * i * i + 5 * i + 2) * c - 8 * i * (7 * i * i + 1) * b + 22 * i * (i - 1) * (2 * i - 1) * a
    break if j % ((i + 1) ** 3) > 0
    a, b, c = b, c, j / ((i + 1) ** 3)
    ary << b
  end
  ary[2..-1]
end

a(n) ~ c * d^n / (Pi*n)^(3/2), where d = 16.8275008141470347474718307386716769... is the real root of the equation -44 + 56*d - 20*d^2 + d^3 = 0 and c = 1.83051467150137478416073409831908489312609... is the positive real root of the equation -1331 - 1020*c^2 - 1936*c^4 + 704*c^6 = 0. - Vaclav Kotesovec, Apr 02 2017

cp's OEIS Frontend

A001421 a(n) = (6*n)!/((n!)^3*(3*n)!).

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A184424 a(n) = (3^n/n!^2) * Product_{k=1..n} (6k-4)*(6k-5).

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A184896 a(n) = C(2n,n) * (7^n/n!^2) * Product_{k=0..n-1} (7k+1)*(7k+6).

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A184892 a(n) = C(2n,n) * (5^n/n!^2) * Product_{k=0..n-1} (5k+1)*(5k+4).

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A184898 a(n) = C(2n,n) * (8^n/n!^2) * Product_{k=0..n-1} (8k+1)*(8k+7).

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A284756 (n + 1)^3*a(n + 1) = 2*(2*n + 1)*(5*n^2 + 5*n + 2)*a(n) - 8*n*(7*n^2 + 1)*a(n - 1) + 22*n*(n - 1)*(2*n - 1)*a(n - 2), with a(0) = 1, a(1) = 4 and a(2) = 28.

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A001421 a(n) = (6n)!/((n!)^3(3*n)!).

A284756 (n + 1)^3a(n + 1) = 2(2n + 1)(5n^2 + 5n + 2)a(n) - 8n(7n^2 + 1)a(n - 1) + 22n(n - 1)(2n - 1)a(n - 2), with a(0) = 1, a(1) = 4 and a(2) = 28.