A289061 -id:A289061 - cp's OEIS frontend

A007242 McKay-Thompson series of class 2a for the Monster group.

1, -492, -22590, -367400, -3764865, -28951452, -182474434, -990473160, -4780921725, -20974230680, -84963769662, -321583404672, -1147744866180, -3890805976500, -12601590210180, -39183052547592, -117437602167291, -340431109329600, -957251463332600, -2617490612355240, -6975126788952456, -18149106017123576, -46187557595906250

Offset: 0

N. J. A. Sloane

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A more correct name would be: Expansion of replicable function of class 2a. See Alexander et al., 1992. - N. J. A. Sloane, Jun 12 2015
From "More on Replicable Functions": 'The fifth row consists of the class names. As stated above, the numbers are the replication orders. For those functions arising in Monstrous Moonshine, the letter corresponds to the relevant conjugacy class in the Monster in Atlas notation (or, if there is more than one class, the one with the first letter). For non-monstrous functions, the class names use lower case letters and, in accordance with Atlas notation, are arranged generally in descending order of Frobenian.'
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			T2a = 1/q - 492*q - 22590*q^3 - 367400*q^5 - 3764865*q^7 - ...
196884 - (-492) = 197376 = 256 * 771, 21493760 - 0 = 256 * 83960, ...

T. Gannon, Moonshine Beyond the Monster, Cambridge, 2006; see p. 425.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..5000 (terms 0..500 from G. A. Edgar)
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
D. Alexander, C. Cummins, J. McKay and C. Simons, Completely replicable functions, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for McKay-Thompson series for Monster simple group

(q*(j(q)-1728))^(k/24): A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), this sequence (k=12), A289063 (k=24).

Cf. A000521, A014708, A289061.

a[ n_] :=  If[ n < 1, Boole[n == 0], SeriesCoefficient[ Sqrt[ 1728 (KleinInvariantJ[ Log[x] /(Pi I)] - 1) + O[x]^(2 n)], {x, 0, 2 n - 1}]] (* Michael Somos, Jun 29 2011 *)
nmax = 30; CoefficientList[Series[x^(1/2)*(-8*(2*EllipticTheta[2, 0, Sqrt[x]]^12 - 3*EllipticTheta[2, 0, Sqrt[x]]^8* EllipticTheta[3, 0, Sqrt[x]]^4 - 3*EllipticTheta[3, 0, Sqrt[x]]^8* EllipticTheta[2, 0, Sqrt[x]]^4 + 2*EllipticTheta[3, 0, Sqrt[x]]^12))/(EllipticTheta[3, 0, Sqrt[x]]^4*(EllipticTheta[2, 0, Sqrt[x]]^4 - EllipticTheta[3, 0, Sqrt[x]]^4)* EllipticTheta[2, 0, Sqrt[x]]^4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 11 2017, check of formula by G. A. Edgar *)
eta[q_]:= q^(1/24)*QPochhammer[q]; nmax = 55; f1A := (eta[q]/eta[q^2] )^24*(1 +256*(eta[q^2]/eta[q])^24)^3; A007242:= CoefficientList[ Series[(q*f1A - 1728*q + O[q]^nmax)^(1/2), {q, 0, 50}], q]; Table[ A007242[[n]], {n, 1, 50}] (* G. C. Greubel, May 09 2018 *)

{a(n) = if( n<0, 0, polcoeff( sqrt( ellj( x^2 * (1 + x * O(x^(2*n)) ) ) - 1728), 2*n - 1))} /* Michael Somos, Jun 29 2011 */

{a(n) = if( n<0, 0, polcoeff( sum( k=1, n, -504 * sigma(k, 5) * x^k, 1 + x * O(x^n)) / eta(x + x * O(x^n))^12, n))} /* Michael Somos, Mar 17 2013 */

Sqrt(j-1728), where j is the j-function, see A000521.
A014708(2*n - 1) == a(n) (mod 256). That is, the coefficients of (T1A - T2a) are all divisible by 256. - Michael Somos, Jun 29 2011
Expansion of (-phi(-q)^12 - 30 * phi(-q)^8 * phi(q)^4 + 96 * phi(-q)^4 * phi(q)^8 - 64 * phi(q)^12) / f(-q)^12 where phi(), f() are Ramanujan theta functions. - Michael Somos, Mar 17 2013
Expansion of (-8*(2*theta_2(0, q)^12-3*theta_2(0, q)^8*theta_3(0, q)^4-3*theta_3(0, q)^8*theta_2(0, q)^4+2*theta_3(0, q)^12))/(theta_3(0, q)^4*(theta_2(0, q)^4-theta_3(0, q)^4)*theta_2(0, q)^4) in powers of q.  Shows an analytic choice of the square root for complex q, 0 < |q| < 1. - G. A. Edgar, Mar 10 2017
G.f.: Product_{k>=1} (1-q^k)^(A289061(k)/2). - Seiichi Manyama, Jul 02 2017
a(n) ~ -exp(2*Pi*sqrt(2*n)) / (2^(3/4) * n^(3/4)). - Vaclav Kotesovec, Jul 09 2017

A289063 Coefficients in expansion of E_6^2/Product_{k>=1} (1-q^k)^24.

Original entry on oeis.org

1, -984, 196884, 21493760, 864299970, 20245856256, 333202640600, 4252023300096, 44656994071935, 401490886656000, 3176440229784420, 22567393309593600, 146211911499519294, 874313719685775360, 4872010111798142520, 25497827389410525184, 126142916465781843075

Offset: 0

Seiichi Manyama, Jun 23 2017

sign

Convolution square of A007242. - Michael Somos, Mar 31 2019

			G.f. = (1-q)^984 * (1-q^2)^286752 * (1-q^3)^102360024 * ...
G.f. = 1 - 984*q + 196884*q^2 + 21493760*q^3 + 864299970*q^4 + 20245856256*q^5 + ... .

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
Eric Weisstein's World of Mathematics, j-Function
Wikipedia, j-invariant

Cf. A000594, A013973 (E_6), A289061, A289062.

Cf. A000521 (j), A007242, A014708, A007240.

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^2 / Product[(1 - x^k)^24, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)
a[ n_] := SeriesCoefficient[ q Series[ 1728 (KleinInvariantJ[Log[q] / (2 Pi I)] - 1), {q, 0, n}], {q, 0, n}]; (* Michael Somos, Mar 31 2019 *)

{a(n) = my(A, U1, U2); if( n<0, 0, A = x * O(x^n); U1 = eta(x + A)^24; U2 = eta(x^2 + A)^24; polcoeff( (U1 - 512*x * U2)^2 * (U1 + 64*x * U2) / (U1^2 * U2), n))}; /* Michael Somos, Mar 31 2019 */

G.f.: Product_{k>=1} (1-q^k)^A289061(k).
a(n) = A000521(n-1) for n = 0 and n > 1.
a(n) ~ exp(4*Pi*sqrt(n)) / (sqrt(2) * n^(3/4)). - Vaclav Kotesovec, Jul 09 2017
G.f.: q * (j(q) - 1728) where j(q) is a modular function. - Michael Somos, Mar 31 2019

A106203 Coefficients of ((j(q)-1728)q)^(1/24) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, -41, -11128, -3785793, -1476507895, -618962022329, -271503819749095, -122857395553223337, -56870247894888518054, -26784343611333662213130, -12787694574831980406719382, -6172809198874485994313412898

Offset: 0

Michael Somos, Apr 25 2005

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

(q*(j(q)-1728))^(k/24): A289563 (k=-96), A289562 (k=-72), A289561 (k=-48), A289417 (k=-24), A289416 (k=-1), this sequence (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), A007242 (k=12), A289063 (k=24).

Cf. A000521, A289061.

CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(1/24), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

{a(n)=if(n<0,0, polcoeff( ((ellj(x+x^2*O(x^n))-1728)*x)^(1/24),n))}

G.f.: Product_{k>=1} (1-q^k)^(A289061(k)/24). - Seiichi Manyama, Jul 02 2017

a(n) ~ c * exp(2*Pi*n) / n^(13/12), where c = -2^(1/12) * Pi^(25/12) * exp(-Pi/12) / (3^(13/12) * Gamma(2/3)^2 * Gamma(3/4)^(7/3) * Gamma(1/12)) = -0.0794786705643291777786030631826408355507134016936764993676699378963... - Vaclav Kotesovec, Mar 07 2018

A289334 Coefficients of (q*(j(q)-1728))^(1/4) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, -246, -41553, -10405738, -3425019885, -1274958998550, -510099547824244, -214102720094848884, -92997705562440483771, -41448768067643091078680, -18848488732890018582016056, -8710420728901868885695224690

Offset: 0

Seiichi Manyama, Jul 02 2017

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

(q*(j(q)-1728))^(k/24): A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), this sequence (k=6), A007242 (k=12), A289063 (k=24).

Cf. A289061.

CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(1/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

G.f.: Product_{k>=1} (1-q^k)^(A289061(k)/4).

a(n) ~ c * exp(2*Pi*n) / n^(3/2), where c = -3 * exp(-Pi/2) / (2^(1/2) * Gamma(3/4)^2) = -0.293663850547434552890056440879436571786655817166913678971... - Vaclav Kotesovec, Mar 07 2018

A289331 Coefficients of (q*(j(q)-1728))^(1/8) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, -123, -28341, -8688812, -3182839959, -1275218435124, -539854235696065, -237249494737728429, -107125917871853210346, -49374268015554366062883, -23126111889684391337303994, -10973394463170114841113101133

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

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

(q*(j(q)-1728))^(k/24): A106203 (k=1), A289330 (k=2), this sequence (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), A007242 (k=12), A289063 (k=24).

Cf. A289061.

CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(1/8), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

G.f.: Product_{k>=1} (1-q^k)^(A289061(k)/8).

a(n) ~ c * exp(2*Pi*n) / n^(5/4), where c = -3^(1/2) * Pi^(1/4) * exp(-Pi/4) / (2^(7/4) * Gamma(3/4)^2) = -0.20815359871514720517220474749202446933362532... - Vaclav Kotesovec, Mar 07 2018

A289330 Coefficients of (q*(j(q)-1728))^(1/12) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, -82, -20575, -6659090, -2518748380, -1032593788260, -445059365317243, -198496352611395190, -90757000595281589335, -42287493553947286567980, -19998274348368716713055507, -9571416182750599673509425808

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

(q*(j(q)-1728))^(k/24): A106203 (k=1), this sequence (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), A007242 (k=12), A289063 (k=24).

Cf. A289061.

CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(1/12), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

G.f.: Product_{k>=1} (1-q^k)^(A289061(k)/12).

a(n) ~ c * exp(2*Pi*n) / n^(7/6), where c = -Pi^(2/3) * exp(-Pi/6) / (2^(1/6) * 3^(7/6) * Gamma(2/3)^2 * Gamma(3/4)^(2/3)) = -0.149642588746726354370104662... - Vaclav Kotesovec, Mar 07 2018

A289332 Coefficients of (q*(j(q)-1728))^(1/6) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, -164, -34426, -9943880, -3522075375, -1378091288700, -572783373894746, -247966590624315128, -110550043138808626860, -50393645499572805001180, -23374903983625804137812564, -10995211137216964385513242408

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

(q*(j(q)-1728))^(k/24): A106203 (k=1), A289330 (k=2), A289331 (k=3), this sequence (k=4), A289333 (k=5), A289334 (k=6), A007242 (k=12), A289063 (k=24).

Cf. A289061.

CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(1/6), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

G.f.: Product_{k>=1} (1-q^k)^(A289061(k)/6).

a(n) ~ c * exp(2*Pi*n) / n^(4/3), where c = -2^(1/3) * Pi^(1/3) * exp(-Pi/3) / (3^(1/3) * Gamma(2/3) * Gamma(3/4)^(4/3)) = -0.252847812633789641246665071437... - Vaclav Kotesovec, Mar 07 2018

A289333 Coefficients of (q*(j(q)-1728))^(5/24) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, -205, -38830, -10493215, -3586921610, -1369515719416, -558606292282075, -238153389340570570, -104811899537297598195, -47246821512435762941195, -21700419062680514765163503, -10118052721530705778119535745

Offset: 0

Seiichi Manyama, Jul 02 2017

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(q*(j(q)-1728))^(k/24): A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), this sequence (k=5), A289334 (k=6), A007242 (k=12), A289063 (k=24).

Cf. A289061.

CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(5/24), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

G.f.: Product_{k>=1} (1-q^k)^(5*A289061(k)/24).

a(n) ~ c * exp(2*Pi*n) / n^(17/12), where c = -5 * 3^(1/3) * Gamma(2/3)^2 * exp(-5*Pi/12) * Gamma(1/12) / (2^(49/12) * Pi^(19/12) * Gamma(3/4)^(5/3)) = -0.28184482434015938133067183460309604452260645657140372869996481157015... - Vaclav Kotesovec, Mar 07 2018

A289417 Coefficients of 1/(q*(j(q)-1728)) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 984, 771372, 543802432, 361216628430, 230920762687776, 143732944930479800, 87718753215371355648, 52729710063184125105381, 31319171802847165756090320, 18421996714811488321383528228, 10748837396953435386200311855872

Offset: 0

Seiichi Manyama, Jul 06 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..365

(q*(j(q)-1728))^(k/24): A289563 (k=-96), A289562 (k=-72), A289561 (k=-48), this sequence (k=-24), A289416 (k=-1), A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), A007242 (k=12), A289063 (k=24).

Cf. A289061, A289209.

CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(-1), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

G.f.: Product_{n>=1} (1-q^n)^(-A289061(n)).

a(n) ~ c * exp(2*Pi*n) * n, where c = Gamma(3/4)^8 * exp(2*Pi) / (324 * Pi^2) = 0.851487576721136974981670736748581778120097667011853803210435262759745... - Vaclav Kotesovec, Mar 07 2018

A289416 Coefficients of (q*(j(q)-1728))^(-1/24) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 41, 12809, 4767210, 1969719570, 861799083811, 391094324350380, 182038077972154741, 86322373755372340110, 41521193849940130872000, 20197774625594843441436930, 9915082544034345319047507780

Offset: 0

Seiichi Manyama, Jul 06 2017

nonn

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

(q*(j(q)-1728))^(k/24): A289563 (k=-96), A289562 (k=-72), A289561 (k=-48), A289417 (k=-24), this sequence (k=-1), A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), A007242 (k=12), A289063 (k=24).

CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(-1/24), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

G.f.: Product_{n>=1} (1-q^n)^(-A289061(n)/24) = Product_{n>=1} (1-q^n)^(1-A289396(n)).

a(n) ~ c * exp(2*Pi*n) / n^(11/12), where c = Gamma(3/4)^(1/3) * exp(Pi/12) / (2^(1/12) * 3^(1/6) * Pi^(1/12) * Gamma(1/12)) = 0.086380262154841817375196725... - Vaclav Kotesovec, Mar 07 2018

cp's OEIS Frontend

A007242 McKay-Thompson series of class 2a for the Monster group.

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A289063 Coefficients in expansion of E_6^2/Product_{k>=1} (1-q^k)^24.

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A106203 Coefficients of ((j(q)-1728)q)^(1/24) where j(q) is the elliptic modular invariant.

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A289334 Coefficients of (q*(j(q)-1728))^(1/4) where j(q) is the elliptic modular invariant.

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A289331 Coefficients of (q*(j(q)-1728))^(1/8) where j(q) is the elliptic modular invariant.

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A289330 Coefficients of (q*(j(q)-1728))^(1/12) where j(q) is the elliptic modular invariant.

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A289332 Coefficients of (q*(j(q)-1728))^(1/6) where j(q) is the elliptic modular invariant.

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A289333 Coefficients of (q*(j(q)-1728))^(5/24) where j(q) is the elliptic modular invariant.

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A289417 Coefficients of 1/(q*(j(q)-1728)) where j(q) is the elliptic modular invariant.