A305757 -id:A305757 - cp's OEIS frontend

A192731 Euler transform is 1 / (q j(q)) where j is j-function (A000521).

-744, 80256, -12288744, 2126816256, -392642298600, 75506620496256, -14935073808384744, 3015675387953504256, -618587635244888064744, 128473308888136855075200, -26951900214112779571200744

Offset: 1

Michael Somos, Jul 08 2011

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			From _Seiichi Manyama_, Jun 18 2017: (Start)
a(1) = (1/1) * A008683(1/1) * A288261(1) = (1/1) * (-744) = -744,
a(2) = (1/2) * (A008683(2/1) * A288261(1) + A008683(2/2) * A288261(2)) = (1/2) * (744 + 159768) = 80256. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..424
B. Brent, p-adic continuity for exponents in product decomposition of the j-invariant, Answer 3 by W. Zudilin
N. J. A. Sloane, Transforms

Cf. A008683, A063995, A110163, A192732, A288261, A302407, A305757.

{a(n) = local(A, S); if( n<1, 0, A = 1 + x * O(x^n); S = x * ellj( x * A ); for( k = 1, n-1, S *= (A - x^k) ^ polcoeff( S, k)); - polcoeff( S, n))}

1 / (q j(q)) = Product_{k>0} (1 - x^k)^-a(k).
a(n) = 3*(A110163(n) - 8) = (1/n) * Sum_{d|n} A008683(n/d) * A288261(d). - Seiichi Manyama, Jun 18 2017
a(n) ~ (-1)^n * 3*exp(Pi*sqrt(3)*n) / n. - Vaclav Kotesovec, Mar 24 2018

A305756 Coefficients of (q*(j(q)-720))^(1/24) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 1, 8192, 707073, -754075135, -132208502783, 90102565204481, 25124693308972545, -11606164284986636798, -4751761734938773786110, 1495856955988144882193922, 890018844816101689979518466, -181104153998957724140261556733

Offset: 0

Seiichi Manyama, Jun 10 2018

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(Conjecture)
Let |b| = 2^p * 3^q * 5^r * ... .
And f(0) = 24, f(b) = 2^(max(0, min(3, p - 1))) * 3^(max(0, min(1, q - 1))) for |b|>0. (See A305762)
Coefficients of (q*(j(q)+b))^(1/f(b)) are integers.
Especially, coefficients of (q*(j(q)+144*k))^(1/24) are integers.
In case of b = -744, |b| = 2^3 * 3^1 * 31 and f(b) = 4. So coefficients of (q*(j(q)-744))^(1/4) are integers. (See A304020)

Seiichi Manyama, Table of n, a(n) for n = 0..378

(q*(j(q)+144*k))^(1/24): A106205 (k=0), this sequence (k=-5), A106203 (k=-12).
(q*(j(q)-720))^(m/24): A305760 (m=-24), A305758 (m=-1), this sequence (m=1).
Cf. A000521, A007240 (j(q)-720), A304020, A305757, A305762.

A305758 Coefficients of (q*(j(q)-720))^(-1/24) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, -1, -8191, -690690, 822573570, 142081621501, -102803639194620, -28281091311058949, 13668272700177314310, 5514776417237892113400, -1808663163639294990384630, -1056880191873167904305607180, 224746784451252089897697350159

Offset: 0

Seiichi Manyama, Jun 10 2018

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Seiichi Manyama, Table of n, a(n) for n = 0..378

(q*(j(q)-720))^(m/24): A305760 (m=-24), this sequence (m=-1), A305756 (m=1).

Cf. A000521, A007240 (j(q)-720), A305757.

Convolution inverse of A305756.

A305760 Coefficients of 1/(q*(j(q)-720)) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, -24, -196308, -12057152, 38590826190, 5667574866912, -7304962792606024, -1755598494902269440, 1325502689549152990437, 465173370338426065214640, -228213884020015849568089308, -112934890287321570650976240384

Offset: 0

Seiichi Manyama, Jun 10 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..376

(q*(j(q)-720))^(m/24): this sequence (m=-24), A305758 (m=-1), A305756 (m=1).

Cf. A000521, A007240 (j(q)-720), A305699, A305757.

G.f.: Product_{k>0} (1 - x^k)^A305757(k).

cp's OEIS Frontend

A192731 Euler transform is 1 / (q j(q)) where j is j-function (A000521).

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A305756 Coefficients of (q*(j(q)-720))^(1/24) where j(q) is the elliptic modular invariant.

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A305758 Coefficients of (q*(j(q)-720))^(-1/24) where j(q) is the elliptic modular invariant.

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A305760 Coefficients of 1/(q*(j(q)-720)) where j(q) is the elliptic modular invariant.

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