A289292 -id:A289292 - cp's OEIS frontend

A108091 Coefficients of series whose 8th power is the theta series of E_8 (see A004009).

1, 30, -2880, 416640, -69178110, 12378401280, -2321610157440, 449733567736320, -89200812128140800, 18013245273252679710, -3689479088922151082880, 764375901202388789804160, -159862757100127037505991680, 33699694000689939789618455040, -7152050326608893289997995966720, 1526705794390267864554876727856640

Offset: 0

N. J. A. Sloane and Michael Somos, Jun 06 2005

sign

			More precisely, the theta series of E_8 begins 1 + 240*q^2 + 2160*q^4 + 6720*q^6 + 17520*q^8 + ... and the 8th root of this is 1 + 30*q^2 - 2880*q^4 + 416640*q^6 - 69178110*q^8 + ...

N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

Seiichi Manyama, Table of n, a(n) for n = 0..424
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
N. J. A. Sloane, Seven Staggering Sequences.

E_4^(k/8): this sequence (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5).

Cf. A001158, A004009 (E_4), A106205, A110163, A300147, A303007.

nmax = 20; s = 8; CoefficientList[Series[(1 - 2*s/BernoulliB[s] * Sum[DivisorSigma[s - 1, k]*x^k, {k, 1, nmax}])^(1/16), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2017 *)

R. = PowerSeriesRing(ZZ,20)
a = R(eisenstein_series_qexp(4,20, normalization='integral'))
list(a.sqrt().sqrt().sqrt()) # Andy Huchala, Jul 10 2021

G.f.: Product_{n>=1} (1-q^n)^(A110163(n)/8). - Seiichi Manyama, Jul 02 2017
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(9/8), where c = 3^(1/4) * Gamma(1/3)^(9/4) / (2^(33/8) * Pi^(3/2) * Gamma(7/8)) = 0.1141392450598624077174159151600898926678394937157356242319309115... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A300147(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 27 2018
G.f.: Sum_{k>=0} A303007(k) * (-f(q))^k where f(q) is Sum_{k>=1} sigma_3(k)*q^k. - Seiichi Manyama, Jun 15 2018

A289293 Coefficients in expansion of E_6^(1/2).

Original entry on oeis.org

1, -252, -40068, -10158624, -3362961924, -1254502939032, -502480721822688, -211053631376919744, -91717692784641665028, -40892713821496126310364, -18600635229558474625901928, -8597703758971125751979122656

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..367

E_k^(1/2): A289291 (k=2), A289292 (k=4), this sequence (k=6), A004009 (k=8), A289294 (k=10), A289295 (k=14).

Cf. A013973 (E_6), A288851.

terms = 12;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E6[x]^(1/2) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

G.f.: Product_{n>=1} (1-q^n)^(A288851(n)/2).

a(n) ~ c * exp(2*Pi*n) / n^(3/2), where c = -3*sqrt(2)*Pi^(3/2) / (16*Gamma(3/4)^8) = -0.2903826839827320330247215149377503818798115... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018

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

Original entry on oeis.org

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

A289392 Coefficients in expansion of E_2^(1/4).

Original entry on oeis.org

1, -6, -72, -1104, -20238, -405792, -8601840, -189317568, -4281478272, -98841343686, -2318973049008, -55118876238000, -1324194430710912, -32099173821105312, -784045854628721568, -19276683937074656064, -476644852188898489662

Offset: 0

Seiichi Manyama, Jul 05 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..702

E_2^(k/4): this sequence (k=1), A289291 (k=2), A289393 (k=3).
E_k^(1/4): this sequence (k=2), A289307 (k=4), A289326 (k=6), A289292 (k=8), A110150 (k=10), A289391 (k=14).
Cf. A004984, A006352 (E_2), A108091, A109817, A289394.

nmax = 20; CoefficientList[Series[(1 - 24*Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}])^(1/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: Product_{n>=1} (1-q^n)^A289394(n).
a(n) ~ c / (n^(5/4) * r^n), where r = A211342 = 0.03727681029645165815098078565... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24 and c = -0.209452682241344640265132676904094736935029272937832600102950644347... - Vaclav Kotesovec, Jul 08 2017
G.f.: Sum_{k>=0} A004984(k) * (3*f(q))^k where f(q) is Sum_{k>=1} sigma_1(k)*q^k. - Seiichi Manyama, Jun 16 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)

A289291 Coefficients in expansion of E_2^(1/2).

Original entry on oeis.org

1, -12, -108, -1344, -22044, -409752, -8201088, -172293504, -3746915388, -83625518604, -1904468689368, -44079484775616, -1033852665619200, -24518163456010392, -586936016770722048, -14164129272396668544, -344209494372831399036

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..702

E_k^(1/2): this sequence (k=2), A289292 (k=4), A289293 (k=6), A004009 (k=8), A289294 (k=10), A289295 (k=14).

Cf. A006352 (E_2), A288968.

G.f.: Product_{n>=1} (1-q^n)^(A288968(n)/2).

a(n) ~ c / (r^n * n^(3/2)), where r = A211342 = 0.03727681029645165815098078... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24 and c = -0.297340792206337929158904153045493466135450465337136... - Vaclav Kotesovec, Jul 02 2017

A289308 Coefficients in expansion of E_4^(3/8).

Original entry on oeis.org

1, 90, -5940, 758520, -115431930, 19355028840, -3447208777320, 639751846440960, -122326632902618100, 23925871041887048130, -4763590542726586318440, 962102309316632909723880, -196619722885250960565506040, 40580696990507644723354537320

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..424

E_4^(k/8): A108091 (k=1), A289307 (k=2), this sequence (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7).

Cf. A004009 (E_4), A110163.

nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^(3/8), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(3*A110163(n)/8).

a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(11/8), where c = 3^(7/4) * Gamma(1/3)^(27/4) / (64 * 2^(3/8) * Pi^(9/2) * Gamma(5/8)) = 0.2574920621515873836544977885672468081360882154861344422709504189964... - Vaclav Kotesovec, Jul 09 2017, updated Mar 05 2018

A289309 Coefficients in expansion of E_4^(5/8).

Original entry on oeis.org

1, 150, -5400, 625200, -86672550, 13570016400, -2289741037200, 406440122001600, -74830416797043000, 14162747887897808550, -2738995393669565720400, 538973037306449327998800, -107578899914865970323788400, 21729813219122500082762389200

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..425

E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), this sequence (k=5), A289318 (k=6), A289319 (k=7).

Cf. A004009 (E_4), A110163.

nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^(5/8), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(5*A110163(n)/8).

a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(13/8), where c = 5 * 3^(5/4) * Gamma(1/3)^(45/4) / (256 * 2^(5/8) * Pi^(15/2) * Gamma(3/8)) = 0.2571085249207580781634342667473393997795373224370302803101380883544... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

A289319 Coefficients in expansion of E_4^(7/8).

Original entry on oeis.org

1, 210, -1260, 232680, -28907970, 4211355960, -671557897080, 113817372354240, -20151698294479500, 3687092782592216970, -692109989731133096760, 132609267059636375116920, -25838624519733523814390760, 5105657091664960508653858680

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

In general, for 0 < m < 1, the expansion of (E_4)^m is asymptotic to (-1)^(n+1) * m * 3^(2*m) * Gamma(1/3)^(18*m) * exp(Pi*sqrt(3)*n) / (2^(9*m) * Pi^(12*m) * Gamma(1-m) * n^(1+m)). - Vaclav Kotesovec, Mar 05 2018

Seiichi Manyama, Table of n, a(n) for n = 0..425

E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), this sequence (k=7).

Cf. A004009 (E_4), A110163.

nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^(7/8), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(7*A110163(n)/8).

a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(15/8), where c = 7 * 3^(7/4) * Gamma(1/3)^(63/4) / (1024 * 2^(7/8) * Pi^(21/2) * Gamma(1/8)) = 0.1121182787986009012644546699220584282491804117887058146553161217384... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

A289318 Coefficients in expansion of E_4^(3/4).

Original entry on oeis.org

1, 180, -3780, 447840, -59046660, 8921092680, -1463828444640, 253953515257920, -45858209756343300, 8534765953624978260, -1626301691950399586280, 315807346469727624396960, -62284193156782292089690080, 12443904711281870749228431240

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..425

E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), this sequence (k=6), A289319 (k=7).

Cf. A004009 (E_4), A110163.

nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^(3/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(3*A110163(n)/4).

a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(7/4), where c = 3^(5/2) * Gamma(1/3)^(27/2) / (256 * 2^(3/4) * Pi^9 * Gamma(1/4)) = 0.2007048471908800363193160136812560289856774734680572658944418664975... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

cp's OEIS Frontend

A108091 Coefficients of series whose 8th power is the theta series of E_8 (see A004009).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A289293 Coefficients in expansion of E_6^(1/2).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A289392 Coefficients in expansion of E_2^(1/4).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A289291 Coefficients in expansion of E_2^(1/2).

Views

Author

Keywords

Links

Crossrefs

Formula

A289308 Coefficients in expansion of E_4^(3/8).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A289309 Coefficients in expansion of E_4^(5/8).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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