A001421 -id:A001421 - cp's OEIS frontend

A289292 Coefficients in expansion of E_4^(1/2).

1, 120, -6120, 737760, -107249640, 17385063120, -3014720249760, 547287510713280, -102701836021530600, 19762301660609250840, -3878226140959368843120, 773209219953012480001440, -156173318001506652330786720, 31888935085481430265623676560

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..425
R. S. Maier, Nonlinear differential equations satisfied by certain classical modular forms, arXiv:0807.1081 [math.NT], 2008-2010, p. 34 equation (7.29b).

E_k^(1/2): A289291 (k=2), this sequence (k=4), A289293 (k=6), A004009 (k=8), A289294 (k=10), A289295 (k=14).
E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), this sequence (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7).
Cf. A001421, A004009 (E_4), A110163.

terms = 14;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^(1/2) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

G.f.: Product_{n>=1} (1-q^n)^(A110163(n)/2).
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(3/2), where c = 3*Gamma(1/3)^9 / (32*sqrt(2)*Pi^(13/2)) = 0.27646925986847687648926173728588572192308632719... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018
G.f.: 3F2(1/6, 1/2, 5/6; 1, 1; 1728/j) where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 07 2017

A092870 Expansion of Hypergeometric function F(1/12, 5/12; 1; 1728*x) in powers of x.

Original entry on oeis.org

1, 60, 39780, 38454000, 43751038500, 54538294552560, 72081445966966800, 99225259048241726400, 140744828381240373790500, 204278086466816584003782000, 301931182921413583820949947280

Offset: 0

Michael Somos, Mar 08 2004

nonn

Self-convolution yields Sum_{k=0..n} a(n-k)*a(k) = A001421(n). - Paul D. Hanna, Jan 25 2011

Seiichi Manyama, Table of n, a(n) for n = 0..310 (terms 0..100 from Vincenzo Librandi)
R. S. Maier, Nonlinear differential equations satisfied by certain classical modular forms, arXiv:0807.1081 [math.NT], 2008-2010, p. 34 equation (7.29a).

Cf. A001421; variants: A184424, A178529, A184891, A184895, A184897. - Paul D. Hanna, Jan 25 2011

Cf. A289307.

CoefficientList[ Series[ Hypergeometric2F1[ 1/12, 5/12, 1, 1728 x], {x, 0, 10}], x]

{a(n) = local(an); if( n<1, n==0, an = vector(n+1); an[1] = 1; for(k=1, n, an[k+1] = an[k] * 12 * (12*k - 7) * (12*k - 11) / k^2); an[n+1])}

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

G.f.: F(1/12, 5/12; 1; 1728*x). a(n) * n^2 = a(n-1) * 12 * (12*n - 7) * (12*n - 11).
G.f. A(x) = y satisfies 0 = (1728*x^2 - x) * y" + (2592*x - 1) * y' + 60 * y.
a(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 + 60*x + 39780*x^2 + 38454000*x^3 +... with  A(x)^2 = 1 + 120*x + 83160*x^2 + 81681600*x^3 +...+ A184894(n)*x^n +... - Paul D. Hanna, Jan 25 2011
a(n) ~ 1728^n * GAMMA(11/12) * GAMMA(7/12) / (4*Pi^2*n^(3/2)). - Vaclav Kotesovec, Apr 20 2014

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

A145493 Coefficients of certain power series associated with Atkin polynomials (see Kaneko-Zagier reference for precise definition).

Original entry on oeis.org

1, 944, 1054170, 1297994880, 1700941165560, 2326960109485440, 3285120488273369460, 4750462777483659350400, 7000542310802147888540760, 10475220761578770433945887360, 15873347459609903883739895346480, 24308956895577230857746294480107520, 37564030601621705287879755722478648000

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, A145494.

{a(n) = (41*n+77)/2310*(6*n+6)!/((3*n+3)!*n!*(n+2)!^2)} \\ Seiichi Manyama, Aug 19 2018

a(n) = (A145492(n+1) - A001421(n+1))/330. - Seiichi Manyama, Aug 18 2018

a(n) = ((41*n+77)/2310) * (6*n+6)!/((3*n+3)!*n!*((n+2)!)^2). - Seiichi Manyama, Aug 19 2018

More terms from Seiichi Manyama, Aug 18 2018

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

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

A361658 Constant term in the expansion of (1 + x^3 + y^3 + z^3 + 1/(xyz))^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 121, 841, 3361, 10081, 25201, 55441, 194041, 1287001, 7927921, 38438401, 152312161, 516079201, 1627691521, 5745472321, 25999820401, 133086258481, 651284938921, 2860955078521, 11312609403481, 42039298455001, 158864460354601, 658342633033801

Offset: 0

Seiichi Manyama, Mar 19 2023

nonn

Cf. A002426, A361657.

Cf. A001421, A361637.

Table[n!*Sum[1/(k!^3 * (3*k)! * (n-6*k)!), {k, 0, n/6}], {n, 0, 30}] (* Vaclav Kotesovec, Mar 20 2023 *)

a(n) = n!*sum(k=0, n\6, 1/(k!^3*(3*k)!*(n-6*k)!));

a(n) = n! * Sum_{k=0..floor(n/6)} 1/(k!^3 * (3*k)! * (n-6*k)!) = Sum_{k=0..floor(n/6)} binomial(n,6*k) * A001421(k).
From Vaclav Kotesovec, Mar 20 2023: (Start)
Recurrence: (n-4)*(n-2)*n^3*a(n) = (6*n^5 - 45*n^4 + 112*n^3 - 123*n^2 + 68*n - 15)*a(n-1) - 3*(n-1)*(5*n^4 - 40*n^3 + 111*n^2 - 132*n + 59)*a(n-2) + 2*(n-2)*(n-1)*(10*n^3 - 75*n^2 + 181*n - 144)*a(n-3) - (n-3)*(n-2)*(n-1)*(15*n^2 - 90*n + 133)*a(n-4) + 3*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 7)*a(n-5) + 1727*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6).
a(n) ~ (1 + 2*sqrt(3))^(n + 3/2) / (4 * 3^(1/4) * Pi^(3/2) * n^(3/2)). (End)

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

cp's OEIS Frontend

A289292 Coefficients in expansion of E_4^(1/2).

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A092870 Expansion of Hypergeometric function F(1/12, 5/12; 1; 1728*x) in powers of x.

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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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A145493 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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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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A361658 Constant term in the expansion of (1 + x^3 + y^3 + z^3 + 1/(x*y*z))^n.

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A361658 Constant term in the expansion of (1 + x^3 + y^3 + z^3 + 1/(xyz))^n.