A092870 -id:A092870 - cp's OEIS frontend

A289307 Coefficients in expansion of E_4^(1/4) in powers of q.

1, 60, -4860, 660480, -105063420, 18206269560, -3328461434880, 631226199152640, -122944850563477500, 24436796345920143420, -4935178772322020730360, 1009598430837232126725120, -208736157503462405753487360, 43541664791244563211024015480

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

			From _Seiichi Manyama_, Jul 07 2017: (Start)
2F1(1/12, 5/12; 1; 1728/j)
= 1 + (1*5)/(1*1) * 12/j + (1*5*13*17)/(1*1*2*2) * (12/j)^2 + (1*5*13*17*25*29)/(1*1*2*2*3*3) * (12/j)^3 + ...
= 1 + 60/j + 39780/j^2 + 38454000/j^3 + ...
= 1 + 60*q - 44640*q^2 + 21399120*q^3 - ...
           + 39780*q^2 - 59192640*q^3 + ...
                       + 38454000*q^3 - ...
                                      + ...
= 1 + 60*q -  4860*q^2 +   660480*q^3 - ... (End)

Seiichi Manyama, Table of n, a(n) for n = 0..424
M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22. Published in B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001, pp. 771-808, section 2.3. Example 3.
R. S. Maier, Nonlinear differential equations satisfied by certain classical modular forms, arXiv:0807.1081 [math.NT], 2008-2010, p. 34 equation (7.29a).

E_4^(k/8): A108091 (k=1), this sequence (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7).

Cf. A000521 (j), A004009 (E_4), A066395 (1/j), A092870, A110163, A289210.

a[ n_] := SeriesCoefficient[ ComposeSeries[ Series[ Hypergeometric2F1[ 1/12, 5/12, 1, q], {q, 0, n}], q^2 / Series[q^2 KleinInvariantJ[ Log[q]/(2 Pi I)], {q, 0, n}]], {q, 0, n}]; (* Michael Somos, Jun 21 2018 *)

G.f.: Product_{n>=1} (1-q^n)^(A110163(n)/4).
G.f.: 2F1(1/12, 5/12; 1; 1728/j) where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 06 2017 [See also the Kontsevich and Zagier link, where t = 1728/j = 1 - Sum_{k>=0} A289210(k)*q^k, with q = q(z) = exp(2*Pi*I*z), Im(z) > 0. - Wolfdieter Lang, May 27 2018]
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(5/4), where c = sqrt(3) * Gamma(1/3)^(9/2) * Gamma(1/4) / (16 * 2^(3/4) * Pi^4) = 0.201967785736579402060958871696381229013432952780653381728912717635... - Vaclav Kotesovec, Jul 07 2017, updated Mar 04 2018

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

Original entry on oeis.org

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)

A184895 a(n) = (7^n/n!^2) * Product_{k=0..n-1} (14k+1)*(14k+6).

Original entry on oeis.org

1, 42, 22050, 16909900, 15269639700, 15109613875944, 15853342647837688, 17325438750851187600, 19510609713302293636050, 22482485054570487449402900, 26382746561837375612125315092, 31419888802098260334367621118904

Offset: 0

Paul D. Hanna, Jan 25 2011

nonn

			G.f.: A(x) = 1 + 42*x + 22050*x^2 + 16909900*x^3 +...
A(x)^2 = 1 + 84*x + 45864*x^2 + 35672000*x^3 +...+ A184896(n)*x^n +...

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A184896; variants: A184424, A178529, A184891, A092870, A184897.

FullSimplify[Table[2^(2*n) * 7^(3*n) * Gamma[n+1/14] * Gamma[n+3/7] / (Gamma[3/7] * Gamma[1/14] * Gamma[n+1]^2), {n, 0, 15}]] (* Vaclav Kotesovec, Jul 03 2014 *)

{a(n)=(7^n/n!^2)*prod(k=0,n-1,(14*k+1)*(14*k+6))}

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

a(n) ~ 2^(2*n) * 7^(3*n) / (Gamma(3/7) * Gamma(1/14) * n^(3/2)). - Vaclav Kotesovec, Nov 19 2023

A184891 a(n) = (5^n/n!^2) * Product_{k=0..n-1} (10k+1)*(10k+4).

Original entry on oeis.org

1, 20, 3850, 1078000, 355066250, 128107903000, 49001272897500, 19520507080800000, 8012558140822125000, 3365274419145292500000, 1439327869068441602250000, 624739666805574817770000000

Offset: 0

Paul D. Hanna, Jan 25 2011

nonn

			G.f.: A(x) = 1 + 20*x + 3850*x^2 + 1078000*x^3  +...
A(x)^2 = 1 + 40*x + 8100*x^2 + 2310000*x^3  +...+ A184892(n)*x^n +...

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A184892; variants: A184424, A178529, A092870, A184895, A184897.

Table[5^n/(n!)^2 Product[(10k+1)(10k+4),{k,0,n-1}],{n,0,20}] (* Harvey P. Dale, Feb 02 2012 *)
FullSimplify[Table[2^(2*n) * 5^(3*n) * Gamma[n+1/10] * Gamma[n+2/5] / (Gamma[2/5] * Gamma[1/10] * Gamma[n+1]^2), {n, 0, 15}]] (* Vaclav Kotesovec, Jul 03 2014 *)

{a(n)=(5^n/n!^2)*prod(k=0,n-1,(10*k+1)*(10*k+4))}

Self-convolution yields Sum_{k=0..n} a(n-k)*a(k) = A184892(n) where

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

A184897 a(n) = (8^n/n!^2) * Product_{k=0..n-1} (16k+1)*(16k+7).

Original entry on oeis.org

1, 56, 43792, 50098048, 67507119680, 99694514343424, 156121609461801984, 254663020429855285248, 428056704465033002591232, 736257531679856764456919040, 1289628692490437108622739390464

Offset: 0

Paul D. Hanna, Jan 25 2011

nonn

			G.f.: A(x) = 1 + 56*x + 43792*x^2 + 50098048*x^3 +...
A(x)^2 = 1 + 112*x + 90720*x^2 + 105100800*x^3 +...+ A184898(n)*x^n +...

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A184898; variants: A184424, A178529, A184891, A092870, A184895.

FullSimplify[Table[2^(11*n) * Gamma[n+1/16] * Gamma[n+7/16] / (Gamma[n+1]^2 * Gamma[1/16] * Gamma[7/16]), {n, 0, 15}]] (* Vaclav Kotesovec, Jul 03 2014 *)

{a(n)=(8^n/n!^2)*prod(k=0,n-1,(16*k+1)*(16*k+7))}

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

A318174 Expansion of Hypergeometric function F(5/12, 13/12; 2; 1728*x) in powers of x.

Original entry on oeis.org

1, 390, 331500, 355699500, 428760177300, 554472661284360, 751706507941225200, 1054268377387568343000, 1516916483664479584186500, 2226631142488300765641223800, 3321243012135549422030449420080, 5019605916068500831023292873530000, 7670343963284674539098285610205650000

Offset: 0

Seiichi Manyama, Aug 20 2018

nonn

A145492 is the convolution of A092870 and this sequence.

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

F([b/2]+5/12, [(b+1)/2]+1/12; b+1; 1728*x): A092870 (b=0), this sequence (b=1), A318200 (b=2), A318201 (b=3).

Cf. A145492.

{a(n) = 12^n/(n!*(n+1)!)*prod(k=0, n-1, (12*k+5)*(12*k+13))}

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

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

A318201 Expansion of Hypergeometric function F(17/12, 25/12; 4; 1728*x) in powers of x.

Original entry on oeis.org

1, 1275, 1641690, 2198770140, 3046553083980, 4336768315045530, 6307588582660665300, 9334870668704489748840, 14013762435241053769769940, 21290019308561214243784932180, 32671991169676632627962261307000, 50573696461217634323724960067290000, 78871365421150941315659866056940998000

Offset: 0

Seiichi Manyama, Aug 21 2018

nonn

A145494 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), A318200 (b=2), this sequence (b=3).

Cf. A145494.

{a(n) = 6*12^n/(n!*(n+3)!)*prod(k=0, n-1, (12*k+17)*(12*k+25))}

a(n) = (6*12^n/(n!*(n+3)!)) * Product_{k=0..n-1} (12k+17)*(12k+25).
a(n) = 6*(12*n+1)*(12*n+5)*(12*n+13)*A092870(n)/(65*(n+1)*(n+2)*(n+3)).
a(n) ~ 2^(6*n + 7) * 3^(3*n + 4) / (65 * Gamma(1/12) * Gamma(5/12) * n^(3/2)). - Vaclav Kotesovec, Aug 21 2018

A289557 Expansion of Hypergeometric function F(1/12, 7/12; 1; 1728*x) in powers of x.

Original entry on oeis.org

1, 84, 62244, 64318800, 76748408100, 99281740718160, 135254824771706640, 191023977418391557440, 277044462249611005649700, 410066847753461267769800400, 616822552390756438979333761680, 940037569843512813004504652800320

Offset: 0

Seiichi Manyama, Jul 07 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..309
R. S. Maier, Nonlinear differential equations satisfied by certain classical modular forms, arXiv:0807.1081 [math.NT], 2008-2010, p. 34 equation (7.30).

Cf. A092870, A289325.

a(n) = (12^n/n!^2) * prod(k=0, n-1, (12*k+1)*(12*k+7)); \\ Michel Marcus, Jul 08 2017

a(n) * n^2 = a(n-1) * 12 * (12*n - 5) * (12*n - 11).
a(n) = (12^n/n!^2) * Product_{k=0..n-1} (12k+1)*(12k+7).
a(n) ~ 2^(6*n-5/6) * 3^(3*n) / (sqrt(Pi) * Gamma(1/6) * n^(4/3)). - Vaclav Kotesovec, Jul 08 2017

cp's OEIS Frontend

A289307 Coefficients in expansion of E_4^(1/4) in powers of q.

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

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A184895 a(n) = (7^n/n!^2) * Product_{k=0..n-1} (14k+1)*(14k+6).

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A184891 a(n) = (5^n/n!^2) * Product_{k=0..n-1} (10k+1)*(10k+4).

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A184897 a(n) = (8^n/n!^2) * Product_{k=0..n-1} (16k+1)*(16k+7).

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A318174 Expansion of Hypergeometric function F(5/12, 13/12; 2; 1728*x) in powers of x.

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

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A318201 Expansion of Hypergeometric function F(17/12, 25/12; 4; 1728*x) in powers of x.

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