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

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

Offset: 0

Author

Peter Bala, Nov 07 2024

Keywords

• sign
• easy

Comments

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.

Examples

			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.
		

Crossrefs

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

Programs

Maple

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);