A380556 E.g.f. A(x) satisfies A(x) = real( 1 + x*A(x)^(2*i) ), where i^2 = -1.

1, 1, 0, -12, 48, 820, -14160, -69160, 5900160, -44796960, -3089865600, 88646729600, 1412786918400, -135956951062400, 1023512450688000, 203887248898944000, -7307555382586368000, -252816835499795840000, 26110132266648748032000, -95216226972043640320000, -80962066973581160140800000

Offset

0,4

Links

Table of n, a(n) for n=0..20.

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.

(1) A(x) = real( 1 + x*A(x)^(2*i) ), where i^2 = -1.

(2) A(x) = 1 + x*cos( log( A(x)^2 ) ).

(3) A(x) = exp( L(x) ), where L(x) = Series_Reversion( (exp(x) - 1)/cos(2*x) ) is the e.g.f. of A380555.

(4) A( (exp(x) - 1)/cos(2*x) ) = exp(x).

Example

E.g.f.: A(x) = 1 + x - 12*x^3/3! + 48*x^4/4! + 820*x^5/5! - 14160*x^6/6! - 69160*x^7/7! + 5900160*x^8/8! - 44796960*x^9/9! - 3089865600*x^10/10! + ...
 where A(x) = 1 + x*cos( log( A(x)^2 ) ).
RELATED SERIES.
A(x)^(2*i) = 1 + 2*i*x + (-4 - 2*i)*x^2/2! + (12 - 28*i)*x^3/3! + (164 + 228*i)*x^4/4! + (-2360 + 1440*i)*x^5/5! + (-9880 - 51480*i)*x^6/6! + (737520 + 129440*i)*x^7/7! + (-4977440 + 17821440*i)*x^8/8! + (-308986560 - 306791360*i)*x^9/9! + ...
 where A(x)^(2*i) = cos( log(A(x)^2) ) + i*sin( log(A(x)^2) ).
L(x) = log(A(x)) = x - x^2 - 10*x^3 + 90*x^4 + 364*x^5 - 17760*x^6 + 85280*x^7 + 5447120*x^8 + ... + A380555(n)*x^n/n! + ...
 where L( (exp(x) - 1)/cos(2*x) ) = x.
cos(2*L(x)) = 1 - 4*x^2/2! + 12*x^3/3! + 164*x^4/4! - 2360*x^5/5! - 9880*x^6/6! + 737520*x^7/7! - 4977440*x^8/8! + ... + (a(n+1)/(n+1))*x^n/n! + ...
 where A(x) = 1 + x*cos(2*L(x)).

Mathematica

CoefficientList[Exp[InverseSeries[Series[ (Exp[x] - 1)/Cos[2*x] , {x, 0, 20}], x]], x]Range[0, 20]! (* Stefano Spezia, Jan 29 2025 *)

Prog

(PARI) {a(n) = my(A = 1+x+x*O(x^n)); for(i=1, n, A = real( 1 + x*A^(2*I) +x*O(x^n) )); n!*polcoef(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A380057, A380555.

Sequence in context: A222966 A376185 A077612 * A041272 A022282 A009959

Adjacent sequences: A380553 A380554 A380555 * A380557 A380558 A380559

Keyword

sign

Author

Paul D. Hanna, Jan 28 2025

Status

approved