A109817 -id:A109817 - cp's OEIS frontend

A289540 Coefficients in expansion of 1/E_6^(1/12).

1, 42, 12852, 4780104, 1974512526, 863778376440, 391960077239304, 182430901827757632, 86505196617272556900, 41607881477457256661154, 20239469012268054187498440, 9935363620927698868439915544, 4914082482014906612773260362232

Offset: 0

Seiichi Manyama, Jul 15 2017

nonn

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

E_6^(k/12): A289570 (k=-18), A000706 (k=-12), A289567 (k=-6), this sequence (k=-1), A109817 (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).

Cf. A013973, A288851, A299503.

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(-1/12), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 26 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(-A288851(n)/12).
a(n) ~ c * exp(2*Pi*n) / n^(11/12), where c = 2^(5/12) * Gamma(3/4)^(4/3) / (3^(1/6) * Pi^(1/3) * Gamma(1/12)) = 0.08654217651555778130817946575840803466... - Vaclav Kotesovec, Jul 26 2017, updated Mar 05 2018
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A299503(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 27 2018

A377223 Coefficients of the series whose 12th power is 1/x * series_reversion(x*E_6(x)), where E_6(x) is the Eisenstein series of weight 6.

Original entry on oeis.org

1, 42, 34020, 39770808, 54603156174, 82058923220904, 130685055490645992, 216707827984305135744, 370213729923354622242084, 647073665508052293475274898, 1151627718366568095339000345192, 2079918757332503030219456972007720, 3802403760868562402170776739039126584, 7022808067106759130277006634854345528104

Offset: 0

Peter Bala, Nov 08 2024

Let R = 1 + x*Z[[x]] denote the set of integral power series with constant term equal to 1. Let P_n = {g^n, g in R}. If f belongs to P_n then the power series 1/x * series_reversion(x*f(x)) is also in P_n. Apply Bala, Theorem 1, Corollary 2.

Here we take f to be the Eisenstein series E_6. See A013973. It is known that the 12th root f^(1/12) has integer coefficients (Heninger et al.). See A109817. It follows that the present sequence is integral.

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. A013973, A109817, A377221, A377222.

with(numtheory):
Order := 30:
E_6 := 1 - 504*add(sigma[5](n)*x^n, n = 1..30):
solve(series(x*E_6, x) = y, x):
seq(coeftayl(series((%/y)^(1/12), y), y = 0, n), n = 0..20);

G.f.: A(x) = the 12-fold iterate I^12( 1/E_6(x)^(1/12) ), where the operator I : R -> R is defined by I(f(x)) = 1/x * series_reversion(x/f(x)), showing that the g.f. A(x) is integral.

A299503 a(n) = (1/12) * Sum_{d|n} d * A288851(d).

Original entry on oeis.org

42, 23940, 12795048, 6852216840, 3669291602172, 1964875343509008, 1052174343447263568, 563430581238674063376, 301712374716950167413282, 161564459029576395778765080, 86516419639708839110100858360, 46328820782943003562067180265504

Offset: 1

Seiichi Manyama, Feb 26 2018

nonn

Cf. A013973 (E_6), A109817, A288851, A289540, A300025.

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(%);

A377222 Expansion of (1/x) * series_reversion(x*E_6(x)), where E_6(x) is the Eisenstein series of weight 6.

Original entry on oeis.org

1, 504, 524664, 682155936, 993260754360, 1549502199011088, 2532317522698504800, 4279562991330657500736, 7417781163248322999957048, 13114370611008351235424557656, 23557650424885130928376974026832, 42873898555113763448790865162056672, 78885999686148803144416784491001491680

Offset: 0

Peter Bala, Nov 08 2024

The 12th root of the power series E_6(x) has integral coefficients. See A109817. The 12th root of the g.f. of the present sequence also has integral coefficients. See A377223.

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 12th root of the g.f. A(x)^(1/12) = (1 + 504*x +  524664*x^2 + 682155936*x^3 + 993260754360*x^4 + 1549502199011088*x^5 + 2532317522698504800*x^6 + ...)^(1/12) = 1 + 42*x + 34020*x^2 + 39770808*x^3 + 54603156174*x^4 + 82058923220904*x^5 + 130685055490645992*x^6 + ... lies in Z[[x]].

Cf. A109817, A377220, A377223.

with(numtheory):
Order := 30:
E_6 := 1 - 504*add(sigma[5](n)*x^n, n = 1..30):
solve(series(x*E_6, x) = y, x):
seq(coeftayl(series((%/y), y), y = 0, n), n = 0..20);

cp's OEIS Frontend

A289540 Coefficients in expansion of 1/E_6^(1/12).

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A377223 Coefficients of the series whose 12th power is 1/x * series_reversion(x*E_6(x)), where E_6(x) is the Eisenstein series of weight 6.

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A299503 a(n) = (1/12) * Sum_{d|n} d * A288851(d).

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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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A377222 Expansion of (1/x) * series_reversion(x*E_6(x)), where E_6(x) is the Eisenstein series of weight 6.

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Maple

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.