A006352 -id:A006352 - cp's OEIS frontend

A341872 Coefficients of the series whose 72nd power equals E_2(x)^3/E_6(x), where E_2(x) and E_6(x) are the Eisenstein series A006352 and A013973.

1, 6, 1998, 722484, 291762942, 125454173544, 56146411655460, 25832836404319152, 12128921727745915062, 5783583949613172902394, 2791762868052719757442008, 1360988846025232489401029220, 668925190887642335984231235348, 331039288912491308442251418152952

Offset: 0

Peter Bala, Feb 22 2021

It is easy to see that E_2(x)^3/E_6(x) == 1 - 72*Sum_{k >= 1} (k - 7*k^5)*x^k/(1 - x^k) (mod 432), and also that the integer k - 7*k^5 is always divisible by 6. Hence, E_2(x)^3/E_6(x) == 1 (mod 432). It follows from Heninger et al., p. 3, Corollary 2, that the series expansion of (E_2(x)^2/E_6(x))^(1/72) = 1 + 6*x + 1998*x^2 + 722484*x^3 + 291762942* x^4 + ... has integer coefficients.

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.
Wikipedia, Eisenstein series

Cf. A006352, A013973, A341871 - A341875.

E(2,x) := 1 -  24*add(k*x^k/(1-x^k),   k = 1..20):
E(6,x) := 1 - 504*add(k^5*x^k/(1-x^k), k = 1..20):
with(gfun): series((E(2,x)^3/E(6,x))^(1/72), x, 20):
seriestolist(%);

a(n) ~ c * exp(2*Pi*n) / n^(71/72), where c = 0.013960369132490470055158573616810629626490780934389076244815126342923645628... - Vaclav Kotesovec, Mar 08 2021

A341873 Coefficients of the series whose 24th power equals E_2(x)^5/E_10(x), where E_2(x) and E_10(x) are the Eisenstein series A006352 and A013974.

Original entry on oeis.org

1, 6, 7038, 2002644, 922569342, 380737463400, 175255606306116, 80315525064955440, 38028486993289854966, 18171889608389845598586, 8807723964899085718419480, 4305311468773791666900669828, 2122088430918938935321961736084

Offset: 0

Peter Bala, Feb 23 2021

It is easy to see that E_2(x)^5/E_10(x) == 1 - 24*Sum_{k >= 1} (5*k - 11*k^9)*x^k/(1 - x^k) (mod 144), and also that the integer 5*k - 11*k^9 is always divisible by 6. Hence, E_2(x)^5/E_10(x) == 1 (mod 144). It follows from Heninger et al., p. 3, Corollary 2, that the series expansion of (E_2(x)^5/E_10(x))^(1/24) = 1 + 6*x + 7038*x^2 + 2002644*x^3 + 922569342*x^4 + ... has integer coefficients.

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.
Wikipedia, Eisenstein series

Cf. A006352, A013974, A341871 - A341875.

E(2,x)  := 1 -  24*add(k*x^k/(1-x^k),   k = 1..20):
E(10,x) := 1 - 264*add(k^9*x^k/(1-x^k), k = 1..20):
with(gfun): series((E(2,x)^5/E(10,x))^(1/24), x, 20):
seriestolist(%);

A386814 Coefficients in q-expansion of E_2^4 * E_6, where E_2 and E_6 are respectively the Eisenstein series A006352 and A013973.

Original entry on oeis.org

1, -600, 34920, -157920, -23913240, 297457776, 3581091360, -13666238400, -458367407640, -4230394757880, -25457298127632, -118465178148000, -459399324219360, -1550209298287440, -4682236500918720, -12910757263315776, -32979872278342680, -78921341989665840, -178491991660958520

Offset: 0

Vaclav Kotesovec, Aug 03 2025

sign

Cf. A006352, A013973, A386785.

terms = 20;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
CoefficientList[Series[E2[x]^4*E6[x], {x, 0, terms}], x]

A386815 Coefficients in q-expansion of E_2^4 * E_4^2, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 384, 19008, -3408384, 86384832, 390216960, -20773815552, -154767455232, 1360271378880, 30429758560128, 278226995437440, 1749537534970368, 8664534035259648, 36062711146189056, 131104383085776384, 427185615341306880, 1270776436150340544, 3499300888293305088, 9016032242401655616

Offset: 0

Vaclav Kotesovec, Aug 03 2025

sign

Cf. A006352, A004009, A386787.

terms = 20;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
CoefficientList[Series[E2[x]^4*E4[x]^2, {x, 0, 20}], x]

A386816 Coefficients in q-expansion of E_2^2 * E_4^3, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 672, 145152, 8663424, -337036224, -6505531200, 40579467264, 1996981485312, 25931378854080, 210242562994464, 1273050737441280, 6245511315490944, 26057670474216192, 95466371280176064, 314217417062264832, 945050326572360960, 2631525623493208512, 6854684254893824832

Offset: 0

Vaclav Kotesovec, Aug 03 2025

sign

Cf. A006352, A004009, A386787.

terms = 20;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
CoefficientList[Series[E2[x]^2*E4[x]^3, {x, 0, 20}], x]

A386817 Coefficients in q-expansion of E_2^3 * E_4 * E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

1, -336, -114912, 4151616, 100931712, -2848456800, -37865826432, 222362076288, 7928555745600, 86986313152368, 620751040620480, 3392046804500928, 15293330001535488, 59435665658243616, 204976008706800384, 640351567531186560, 1840291945275505344, 4923361835292283488

Offset: 0

Vaclav Kotesovec, Aug 03 2025

sign

Cf. A006352, A004009, A013973, A386787.

terms = 20;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
CoefficientList[Series[E2[x]^3*E4[x]*E6[x], {x, 0, 20}], x]

A386818 Coefficients in q-expansion of E_2^2 * E_6^2, where E_2 and E_6 are respectively the Eisenstein series A006352 and A013973.

Original entry on oeis.org

1, -1056, 269568, 5490816, -301315008, -6705063360, 41022885888, 1997915006208, 25923296790720, 210257663162208, 1273067731422720, 6245405396604288, 26057761857270528, 95466552284986176, 314217210391363584, 945049912933328640, 2631525397984618944, 6854687219510589888

Offset: 0

Vaclav Kotesovec, Aug 03 2025

sign

Cf. A006352, A013973, A386787.

terms = 20;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
CoefficientList[Series[E2[x]^2*E6[x]^2, {x, 0, 20}], x]

A143278 Convolution of A006352 and A010815.

Original entry on oeis.org

1, -25, -49, 0, 0, 121, 0, 169, 0, 0, 0, 0, -289, 0, 0, -361, 0, 0, 0, 0, 0, 0, 529, 0, 0, 0, 625, 0, 0, 0, 0, 0, 0, 0, 0, -841, 0, 0, 0, 0, -961, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1225, 0, 0, 0, 0, 0, 1369, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1681, 0, 0, 0, 0, 0, 0, -1849, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Aug 04 2008

sign

			q - 25*q^25 - 49*q^49 + 121*q^121 + 169*q^169 - 289*q^289 - 361*q^361 + ...

S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 188

{a(n) = local(m); if(issquare(n = 24*n+1, &m), n * kronecker(12, m))}

Expansion of product of f(-q) * P(q) where f(), P() are Ramanujan series.
G.f.: Sum_{k} (-1)^k * (6*k - 1)^2 * x^(k * (3*k - 1) / 2).
G.f.: (Sum_{k} (-1)^k * x^(k * (3*k - 1) / 2)) * (1 - 24 * Sum_{k>0} k * x^k / (1 -x^k)).

A282017 Coefficients in q-expansion of (E_4 + E_2^2)/2, where E_2 and E_4 are the Eisenstein series shown in A006352 and A004009, respectively.

Original entry on oeis.org

1, 96, 1296, 4992, 13488, 25920, 50112, 74496, 123120, 164832, 246240, 300672, 442176, 501312, 694656, 794880, 1052016, 1135296, 1534032, 1591680, 2086560, 2214912, 2763072, 2840832, 3723840, 3668640, 4590432, 4750080, 5801088, 5728320, 7309440, 7007232, 8697456, 8722944, 10349856, 10160640

Offset: 0

N. J. A. Sloane, Feb 05 2017

nonn

Cf. A004009 and A006352.

with(numtheory); M:=100;
E := proc(k) local n, t1; global M;
t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..M+1);
series(t1, q, M+1); end;
e2:=E(2); e4:=E(4); e6:=E(6);
series((e2^2+e4)/2,q,M+1);
seriestolist(%);

A341842 Coefficients of the series whose 12th power equals E_2*E_4, where E_2 and E_4 are the Eisenstein series shown in A006352 and A004009.

Original entry on oeis.org

1, 18, -2088, 301296, -50784174, 9174627360, -1734603719472, 338286925650240, -67486440186470016, 13697820033167444178, -2818359890320927630320, 586296297186462310481424, -123077156275866375661524864, 26034142700316716015964656544

Offset: 0

Peter Bala, Feb 21 2021

The g.f. is the 12th root of the g.f. of A282019.

It is easy to see that E_2(x)*E_4(x) == 1 - 24*Sum_{k >= 1} (k - 10*k^3)*x^k/(1 - x^k) (mod 72), and also that the integer k - 10*k^3 is always divisible by 3. Hence, E_2(x)*E_4(x) == 1 (mod 72). It follows from Heninger et al., p. 3, Corollary 2, that the series expansion of (E_2(x)*E_4(x))^(1/12) = 1 + 18*x - 2088*x^2 + 301296*x^3 - 50784174*x^4 + ... has integer coefficients.

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; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
Wikipedia, Eisenstein series

A004009, A006352, A108091, A282019, A289392.

E(2,x) := 1 -  24*add(k*x^k/(1-x^k),   k = 1..20):
E(4,x) := 1 + 240*add(k^3*x^k/(1-x^k), k = 1..20):
with(gfun): series((E(2,x)*E(4,x))^(1/12), x, 20):
seriestolist(%);

cp's OEIS Frontend

A341872 Coefficients of the series whose 72nd power equals E_2(x)^3/E_6(x), where E_2(x) and E_6(x) are the Eisenstein series A006352 and A013973.

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A341873 Coefficients of the series whose 24th power equals E_2(x)^5/E_10(x), where E_2(x) and E_10(x) are the Eisenstein series A006352 and A013974.

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A386814 Coefficients in q-expansion of E_2^4 * E_6, where E_2 and E_6 are respectively the Eisenstein series A006352 and A013973.

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A386815 Coefficients in q-expansion of E_2^4 * E_4^2, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

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A386816 Coefficients in q-expansion of E_2^2 * E_4^3, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

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A386817 Coefficients in q-expansion of E_2^3 * E_4 * E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A386818 Coefficients in q-expansion of E_2^2 * E_6^2, where E_2 and E_6 are respectively the Eisenstein series A006352 and A013973.

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A143278 Convolution of A006352 and A010815.

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A282017 Coefficients in q-expansion of (E_4 + E_2^2)/2, where E_2 and E_4 are the Eisenstein series shown in A006352 and A004009, respectively.

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A341842 Coefficients of the series whose 12th power equals E_2*E_4, where E_2 and E_4 are the Eisenstein series shown in A006352 and A004009.

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Maple