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
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.
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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
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.
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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);
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
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]].
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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);
Showing 1-3 of 3 results.
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