cp's OEIS Frontend

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

%I A377220 #16 Nov 17 2024 07:21:32
%S A377220 1,-240,113040,-66534720,43859560080,-30976854078240,
%T A377220 22919806575299520,-17536455012714130560,13761543459443537811600,
%U A377220 -11015192093055645841813680,8958361831335008460574345440,-7381454927286057227098811282880,6148958599311807793865548969813440,-5169975617288319668409172392988655520
%N A377220 Expansion of (1/x) * series_reversion(x*E_4(x)), where E_4(x) denotes the Eisenstein series of weight 4 (see A004009).
%C A377220 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.
%C A377220 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.
%e A377220 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.
%p A377220 with(numtheory):
%p A377220 Order := 30:
%p A377220 E_4 := 1 + 240*add(sigma[3](n)*x^n, n = 1..30):
%p A377220 solve(series(x*E_4, x) = y, x):
%p A377220 seq(coeftayl(series((%/y), y), y = 0, n), n = 0..20);
%Y A377220 Cf. A004009, A108091, A377221, A377222, A377223.
%K A377220 sign,easy
%O A377220 0,2
%A A377220 _Peter Bala_, Nov 07 2024