A145493 -id:A145493 - 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)

A145492 Coefficients of certain power series associated with Atkin polynomials (see Kaneko-Zagier reference for precise definition).

Original entry on oeis.org

1, 450, 394680, 429557700, 522037315800, 678696698599920, 923563866149496000, 1298893924326291064200, 1872892788786285985719000, 2753834730409783196154778800, 4113309164116707723917886096960

Offset: 0

N. J. A. Sloane, Feb 28 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..310
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

Cf. A001421, A145493, A145494, A318174.

{a(n) = (8*n+7)*(6*n+1)!/(7*(3*n)!*n!*(n+1)!^2)} \\ Seiichi Manyama, Aug 20 2018

a(n) = (8*n+7)*(6*n+1)!/(7*(3*n)!*n!*(n+1)!^2). - Seiichi Manyama, Aug 20 2018

More terms from Vladeta Jovovic, Feb 28 2009

A145494 Coefficients of certain power series associated with Atkin polynomials (see Kaneko-Zagier reference for precise definition).

Original entry on oeis.org

1, 1335, 1757970, 2386445040, 3336565609080, 4780478992153590, 6986981484124227300, 10379857332015914783040, 15630336095483779833585240, 23805745537435619756724715080, 36609525021962109091946530420080

Offset: 0

N. J. A. Sloane, Feb 28 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..309
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

Cf. A001421, A145492, A145493.

a(n) = (A145493(n+1) - A145492(n+1))/494. - Seiichi Manyama, Aug 18 2018

More terms from Vladeta Jovovic, Feb 28 2009

A318200 Expansion of Hypergeometric function F(17/12, 13/12; 3; 1728*x) in powers of x.

Original entry on oeis.org

1, 884, 961350, 1166694360, 1514952626460, 2059469884770480, 2894070055573717020, 4170217137221937001200, 6128342594004497520113460, 9149429785497381327907574160, 13838512550564789258460205917000, 21159569553888757349236649959188000, 32653750015126185895018415883446910000

Offset: 0

Seiichi Manyama, Aug 21 2018

nonn

A145493 is the convolution of A092870 and this sequence.

Seiichi Manyama, Table of n, a(n) for n = 0..309
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

F([b/2]+5/12, [(b+1)/2]+1/12; b+1; 1728*x): A092870 (b=0), A318174 (b=1), this sequence (b=2), A318201 (b=3).

Cf. A145493.

{a(n) = 2*12^n/(n!*(n+2)!)*prod(k=0, n-1, (12*k+17)*(12*k+13))}

a(n) = (2*12^n/(n!*(n+2)!)) * Product_{k=0..n-1} (12k+17)*(12k+13).
a(n) = 2*(12*n+1)*(12*n+5)*A092870(n)/(5*(n+1)*(n+2)).
a(n) ~ 2^(6*n + 5) * 3^(3*n + 2) / (5 * Gamma(1/12) * Gamma(5/12) * n^(3/2)). - Vaclav Kotesovec, Aug 21 2018

A145495 Coefficients of certain power series associated with Atkin polynomials (see Kaneko-Zagier reference for precise definition).

Original entry on oeis.org

1, 84, 27720, 13693680, 5354228880, 2489716429200, 1010824870255200, 459492105307435200, 189737418627305920800, 85223723866764909426000, 35532611270849849570013600, 15842376246977818384652245440, 6646596943618421076833646609600, 2948532659526725719238433845966400

Offset: 0

N. J. A. Sloane, Feb 28 2009

nonn

			From _Seiichi Manyama_, Aug 19 2018: (Start)
Phi_0(t)/1       = 1 + 120*t +  83160*t^2 + ... (See A001421).
Phi_1(t)/(84*t)  = 1 + 450*t + 394680*t^2 + ... (See A145492).
Phi_2(t)/(27720*t^2)
= (1 + 450*t + 394680*t^2 + ... - (1 + 120*t +  83160*t^2 + ... ))/(330*t)
= 1 + 944*t + 1054170*t^2 + ... (See A145493).
Phi_3(t)/(13693680*t^3)
= (1 + 944*t + 1054170*t^2 + ... - (1 + 450*t + 394680*t^2 + ... ))/(494*t)
= 1 + 1335*t + 1757970*t^2 + ... (See A145494).
Phi_4(t)/(5354228880*t^4)
= (1 + 1335*t + 1757970*t^2 + ... - (1 + 944*t + 1054170*t^2 + ... ))/(391*t)
= 1 + 1800*t + 2783760*t^2 + ... .
Phi_5(t)/(2489716429200*t^5)
= (1 + 1800*t + 2783760*t^2 + ... - (1 + 1335*t + 1757970*t^2 + ... ))/(465*t)
= 1 + 2206*t + 3863952*t^2 + ... .
Phi_6(t)/(1010824870255200*t^6)
= (1 + 2206*t + 3863952*t^2 + ... - (1 + 1800*t + 2783760*t^2 + ... ))/(406*t)
= 1 + 18624/7*t + 36827541/7*t^2 + ... .
Phi_7(t)/(459492105307435200*t^6)
= (1 + 18624/7*t + 36827541/7*t^2 + ... - (1 + 2206*t + 3863952*t^2 + ... ))/((3182/7)*t)
= 1 + (6147/2)*t + 6715687*t^2 + ... . (End)

Seiichi Manyama, Table of n, a(n) for n = 0..379
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

Cf. A001421, A145492, A145493, A145494.

From Seiichi Manyama, Aug 19 2018: (Start)
a(n) = (6*n+1)!/((n-1)!*(2*n)!*(3*n)!*(6*n+(-1)^n)) for n > 0.
a(n) = 12*(6*n-6+(-1)^(n-1))*(6*n+(-1)^(n-1))*a(n-1)/((n-1)*n) for n > 1. (End)

More terms from Seiichi Manyama, Aug 19 2018

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

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A145492 Coefficients of certain power series associated with Atkin polynomials (see Kaneko-Zagier reference for precise definition).

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A145494 Coefficients of certain power series associated with Atkin polynomials (see Kaneko-Zagier reference for precise definition).

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Author

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Formula

Extensions

A318200 Expansion of Hypergeometric function F(17/12, 13/12; 3; 1728*x) in powers of x.

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PARI

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A145495 Coefficients of certain power series associated with Atkin polynomials (see Kaneko-Zagier reference for precise definition).

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Examples

Links

Crossrefs

Formula

Extensions

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