A380057 E.g.f. satisfies A(x) = real( 1 + x*A(x)^i ), where i^2 = -1.

1, 1, 0, -3, 12, 10, -660, 5600, 8400, -951660, 12715200, 21635900, -4308744000, 80314007800, 115204471200, -44501789202000, 1083368456352000, 782537744170000, -876176569052928000, 26724653123017850000, -10930955906482560000, -29304692085200613900000, 1088420125090964265600000

Offset

0,4

Links

Paul D. Hanna, Table of n, a(n) for n = 0..400

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)^i ), where i^2 = -1.

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

(3) A(x) = exp( L(x) ), where L(x) = Series_Reversion( (exp(x) - 1)/cos(x) ) is the e.g.f. of A380055.

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

(5) A'(x) = A(x) * cos( log(A(x)) ) / (A(x) + x*sin( log(A(x)) )).

Example

E.g.f.: A(x) = 1 + x - 3*x^3/3! + 12*x^4/4! + 10*x^5/5! - 660*x^6/6! + 5600*x^7/7! + 8400*x^8/8! - 951660*x^9/9! + 12715200*x^10/10! + ...
 where A(x) = real( 1 + x*A(x)^i ).
RELATED SERIES.
A(x)^i = 1 + i*x + (-1 - i)*x^2/2! + (3 - 2*i)*x^3/3! + (2 + 24*i)*x^4/4! + (-110 - 90*i)*x^5/5! + (800 - 540*i)*x^6/6! + (1050 + 12640*i)*x^7/7! + (-105740 - 85680*i)*x^8/8! + (1271520 - 808480*i)*x^9/9! + ...
L(x) = log(A(x)) = x - x^2/2! - x^3/3! + 18*x^4/4! - 86*x^5/5! - 210*x^6/6! + 8840*x^7/7! - 80080*x^8/8! - 266220*x^9/9! + ... + A380055(n)*x^n/n! + ...
 where L( (exp(x) - 1)/cos(x) ) = x.
cos(L(x)) = 1 - x^2/2! + 3*x^3/3! + 2*x^4/4! - 110*x^5/5! + 800*x^6/6! + 1050*x^7/7! - 105740*x^8/8! + 1271520*x^9/9! + ... + (a(n+1)/(n+1))*x^n/n! + ...
 where A(x) = 1 + x*cos(L(x)).
sin(L(x)) = x - x^2/2! - 2*x^3/3! + 24*x^4/4! - 90*x^5/5! - 540*x^6/6! + 12640*x^7/7! - 85680*x^8/8! - 808480*x^9/9! + 29636100*x^10/10! + ...
 where A(x)^i = cos(L(x)) + i*sin(L(x)).
(exp(x) - 1)/cos(x) = x + x^2/2! + 4*x^3/3! + 7*x^4/4! + 36*x^5/5! + 91*x^6/6! + 624*x^7/7! + 2087*x^8/8! + 18256*x^9/9! + ... + A380053(n)*x^n/n! + ...
 where A( (exp(x) - 1)/cos(x) ) = exp(x).
SPECIFIC VALUES.
A(t) = 3/2 at t = 0.544117577471602714300218052469...
  where t = (1/2)/real( (3/2)^i ) = (1/2)/cos(log(3/2)).
A(t) = 4/3 at t = 0.347619044845549508650339299345...
  where t = (1/3)/cos(log(4/3)).
A(t) = 5/4 at t = 0.256355931938414599112107620645...
  where t = (1/4)/cos(log(5/4)).
A(t) = 6/5 at t = 0.203370786475336228292002468204...
  where t = (1/5)/cos(log(6/5)).
A(5/9) = 1.509166099041466745516450953830986339123819257989853587...
A(1/2) = 1.464101647089238181480821477030380839753029025795281424...
  where A(1/2) = real(1 + (1/2)*A(1/2)^i) = 1 + (1/2)*cos(log( A(1/2) )).
A(1/3) = 1.320532220599397916493026637544832197337142648617761054...
  where A(1/3) = 1 + (1/3)*cos(log( A(1/3) )).
A(1/4) = 1.244062312309475342914040842626056476267801020397804125...
  where A(1/4) = 1 + (1/4)*cos(log( A(1/4) )).
A(1/5) = 1.196781755871960043253050643301603078428252760913031148...
A(1/6) = 1.164732612936027556202815971272406070730644075186904648...
A(1/10) = 1.099550027586687118480790742605384041459169163472092691...

Prog

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

Crossrefs

Cf. A380053, A380055, A380058.

Sequence in context: A182455 A110345 A018999 * A279305 A217785 A168410

Adjacent sequences: A380052 A380053 A380055 * A380061 A380062 A380063

Keyword

sign,new

Author

Paul D. Hanna, Jan 24 2025

Status

approved