A108091 -id:A108091 - cp's OEIS frontend

A294978 Coefficients in expansion of (E_4/E_2^4)^(1/8).

1, 42, -2268, 395304, -64600914, 11644170552, -2188350306072, 424652412357696, -84326944950450972, 17044476557469661986, -3493525880987663047128, 724189608821718233434296, -151528575864988356484968840, 31955212589107172812017247992

Offset: 0

Seiichi Manyama, Feb 12 2018

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Also coefficients in expansion of (E_8/E_2^8)^(1/16).

Cf. A004009 (E_4), A006352 (E_2), A108091, A289565, A294626, A294974.

terms = 14;
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}];
(E4[x]/E2[x]^4)^(1/8) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Convolution inverse of A294974.
G.f.: Product_{n>=1} (1-q^n)^(-A294626(n)).
a(n) ~ -(-1)^n * Pi^(5/4) * exp(Pi*sqrt(3)*n) / (2^(19/8) * 3^(9/8) * Gamma(2/3)^(9/4) * Gamma(7/8) * n^(9/8)). - Vaclav Kotesovec, Jun 03 2018

A300147 a(n) = (1/8) * Sum_{d|n} d * A110163(d).

Original entry on oeis.org

-30, 6660, -1536120, 354476040, -81800478900, 18876653594640, -4356063194112240, 1005225129672310800, -231970363216834560390, 53530545369975222475800, -12352954264801690636800360, 2850624405442199478575792160

Offset: 1

Seiichi Manyama, Feb 26 2018

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Vaclav Kotesovec, Table of n, a(n) for n = 1..420

Cf. A004009 (E_4), A108091, A110163, A289247, A300025.

a(n) ~ (-1)^n * exp(Pi*sqrt(3)*n) / 8. - Vaclav Kotesovec, Jun 07 2018

A299955 Coefficients in expansion of E_4^(3/2).

Original entry on oeis.org

1, 360, 24840, -465120, 57417480, -6800282640, 930889890720, -139401582644160, 22250341370421000, -3723955494287559480, 646515765251485521840, -115559140273640812421280, 21150946022800731753255840, -3948247836773858791840263120

Offset: 0

Seiichi Manyama, Feb 22 2018

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Seiichi Manyama, Table of n, a(n) for n = 0..426

E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7), A004009 (k=8), this sequence (k=12), A008410 (k=16), A008411 (k=24), A282012 (k=32), A282015 (k=40).

Convolution cube of A289292.

a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(5/2), where c = 81*Gamma(1/3)^27 / (32768*sqrt(2)*Pi^(37/2)) = 0.39832876770813443250501819621900549862424768734... - Vaclav Kotesovec, Mar 05 2018

A341801 Coefficients of the series whose 12th power equals E_2E_4E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973.

Original entry on oeis.org

1, -24, -13932, -3585216, -1580941068, -628142318640, -281617154080704, -126114490533924480, -58596395743623957084, -27537281150571923942424, -13153668428658997172513880, -6345860505664230715931502912, -3091029995619009106117946403456

Offset: 0

Peter Bala, Feb 20 2021

The g.f. is the 12th root of the g.f. of A282102.

It is easy to see that E_2(x)*E_4(x)*E_6(x) == 1 - 24*Sum_{k >= 1} (k - 10*k^3 + 21*k*5)*x^k/(1 - x^k) (mod 72), and also that the integer k - 10*k^3 + 21*k*5 = k*(3*k^2 - 1)*(7^k^2 - 1) is always divisible by 3. Hence, E_2(x)*E_4(x)*E_6(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)* E_6(x))^(1/12) = 1 - 24*x - 13932*x^2 - 3585216*x^3 - 1580941068*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. A004009, A006352, A013973, A108091, A109817, A282102, 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):
E(6,x) := 1 - 504*add(k^5*x^k/(1-x^k), k = 1..20):
with(gfun): series((E(2,x)*E(4,x)*E(6,x))^(1/12), x, 20):
seriestolist(%);

A377220 Expansion of (1/x) * series_reversion(x*E_4(x)), where E_4(x) denotes the Eisenstein series of weight 4 (see A004009).

Original entry on oeis.org

1, -240, 113040, -66534720, 43859560080, -30976854078240, 22919806575299520, -17536455012714130560, 13761543459443537811600, -11015192093055645841813680, 8958361831335008460574345440, -7381454927286057227098811282880, 6148958599311807793865548969813440, -5169975617288319668409172392988655520

Offset: 0

Peter Bala, Nov 07 2024

The 8th root of the power series E_4(x) has integral coefficients. See A108091. The 8th root of the g.f. of the present sequence also has integral coefficients. See A377221.

More generally if f(x) = g(x)^n, where g(x) = 1 + g_1*x + g_2*x^2 + ... is a power series with integral coefficients, then both the power series (1/x) * series_reversion(x*f(x)) and (1/x) * series_reversion(x/f(x)) are also equal to the n-th powers of integral power series.

			The 8th root of the g.f. A(x)^(1/8) = (1 - 240*x + 113040*x^2 - 66534720*x^3 + 43859560080*x^4 - 30976854078240*x^5 + 22919806575299520*x^6 +...)^(1/8) = 1 - 30*x + 10980*x^2 - 5822040*x^3 + 3623245710*x^4 - 2467207358280*x^5 + 1779938570782440*x^6 + .... lies in Z[[x]]. See A377221.

Cf. A004009, A108091, A377221, A377222, A377223.

with(numtheory):
Order := 30:
E_4 := 1 + 240*add(sigma[3](n)*x^n, n = 1..30):
solve(series(x*E_4, x) = y, x):
seq(coeftayl(series((%/y), y), y = 0, n), n = 0..20);

A377221 Coefficients of the series whose 8th power is 1/x * series_reversion(x * E_4(x)), where E_4(x) is the Eisenstein series of weight 4.

Original entry on oeis.org

1, -30, 10980, -5822040, 3623245710, -2467207358280, 1779938570782440, -1336872265001920320, 1034337566576031632100, -818707881037376263396710, 659829780447854309255690280, -539628866179308154664183513160, 446708428717281359928910138018680, -373580804664955058627213489276760840

Offset: 0

Peter Bala, Nov 07 2024

Let R = 1 + x*Z[[x]] denote the set of integer power series with constant term equal to 1. Define the operator T: R -> R by T(f(x)) = 1/x * series_reversion(x*f(x)). Let P_n = {g^n, g in R}. It follows from Bala, Theorem 1, Corollary 2, that if f belongs to P_n then T(f) is also in P_n.

Here we take f to be the Eisenstein series E_4, the theta series of the E_8 lattice. See A004009. It is known that the 8th root E_4^(1/8) has integer coefficients (Heninger et al.). It follows that the present sequence is integral.

			Let F(x) = 1/(E_4(x))^(1/8) = 1 - 30*x + 3780*x^2 - 616440*x^3 + 111056910*x^4 - 21135698280*x^5 + ...
Then
I(F(x))   = 1 - 30*x + 4680*x^2 - 983640*x^3 + 234828510*x^4 - 60324330780*x^5 + ...
I^2(F(x)) = 1 - 30*x + 5580*x^2 - 1431840*x^3 + 422752110*x^4 - 135277163280*x^5 + ...
I^3(F(x)) = 1 - 30*x + 6480*x^2 - 1961040*x^3 + 687787710*x^4 - 262396695780*x^5 + ...
I^4(F(x)) = 1 - 30*x + 7380*x^2 - 2571240*x^3 + 1042895310*x^4 - 461122928280*x^5 + ...
I^5(F(x)) = 1 - 30*x + 8280*x^2 - 3262440*x^3 + 1501034910*x^4 - 753933360780*x^5 + ...
I^6(F(x)) = 1 - 30*x + 9180*x^2 - 4034640*x^3 + 2075166510*x^4 - 1166342993280*x^5 + ...
I^7(F(x)) = 1 - 30*x + 10080*x^2 - 4887840*x^3 + 2778250110*x^4 - 1726904325780*x^5 + ...
I^8(F(x)) = 1 - 30*x + 10980*x^2 - 5822040*x^3 + 3623245710*x^4 - 2467207358280*x^5 + ... = the g.f. A(x).

Peter Bala, Fractional iteration of a series inversion operator
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.

Cf. A004009, A108091, A377220, A377223.

with(numtheory):
Order := 30:
E_4 := 1 + 240*add(sigma[3](n)*x^n, n = 1..30):
solve(series(x*E_4, x) = y, x):
seq(coeftayl(series((%/y)^(1/8), y), y = 0, n), n = 0..20);

G.f.: A(x) = the 8-fold iterate I^8( 1/(E_4(x))^(1/8) ), where I : R -> R denotes the operator I(f(x)) = 1/x * series_reversion(x/f(x)), showing that the g.f. A(x) is integral.

A108772 ( (Theta series of E_8)^(1/8) - (theta series of Leech lattice)^(1/24) ) / 30.

Original entry on oeis.org

0, 1, -369, -9408, 22853743, 4794451200, -3085237931328, -819499646151424, 384739275624398865, 158606464687370095617, -50143660099205286196800, -29592112124024539896414528, 6061193531453485412550560256, 5483327637634568394533944708352

Offset: 0

N. J. A. Sloane and Nadia Heninger, Jun 21 2005

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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.

Equals (A108091-A108093)/30.

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

A294978 Coefficients in expansion of (E_4/E_2^4)^(1/8).

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A300147 a(n) = (1/8) * Sum_{d|n} d * A110163(d).

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A299955 Coefficients in expansion of E_4^(3/2).

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A341801 Coefficients of the series whose 12th power equals E_2*E_4*E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973.

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A377220 Expansion of (1/x) * series_reversion(x*E_4(x)), where E_4(x) denotes the Eisenstein series of weight 4 (see A004009).

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A377221 Coefficients of the series whose 8th power is 1/x * series_reversion(x * E_4(x)), where E_4(x) is the Eisenstein series of weight 4.

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A108772 ( (Theta series of E_8)^(1/8) - (theta series of Leech lattice)^(1/24) ) / 30.

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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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A341801 Coefficients of the series whose 12th power equals E_2E_4E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973.