This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A380057 #20 Jan 25 2025 09:52:23
%S A380057 1,1,0,-3,12,10,-660,5600,8400,-951660,12715200,21635900,-4308744000,
%T A380057 80314007800,115204471200,-44501789202000,1083368456352000,
%U A380057 782537744170000,-876176569052928000,26724653123017850000,-10930955906482560000,-29304692085200613900000,1088420125090964265600000
%N A380057 E.g.f. satisfies A(x) = real( 1 + x*A(x)^i ), where i^2 = -1.
%H A380057 Paul D. Hanna, <a href="/A380057/b380057.txt">Table of n, a(n) for n = 0..400</a>
%F A380057 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
%F A380057 (1) A(x) = real( 1 + x*A(x)^i ), where i^2 = -1.
%F A380057 (2) A(x) = 1 + x*cos( log( A(x) ) ).
%F A380057 (3) A(x) = exp( L(x) ), where L(x) = Series_Reversion( (exp(x) - 1)/cos(x) ) is the e.g.f. of A380055.
%F A380057 (4) A( (exp(x) - 1)/cos(x) ) = exp(x).
%F A380057 (5) A'(x) = A(x) * cos( log(A(x)) ) / (A(x) + x*sin( log(A(x)) )).
%e A380057 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! + ...
%e A380057 where A(x) = real( 1 + x*A(x)^i ).
%e A380057 RELATED SERIES.
%e A380057 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! + ...
%e A380057 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! + ...
%e A380057 where L( (exp(x) - 1)/cos(x) ) = x.
%e A380057 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! + ...
%e A380057 where A(x) = 1 + x*cos(L(x)).
%e A380057 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! + ...
%e A380057 where A(x)^i = cos(L(x)) + i*sin(L(x)).
%e A380057 (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! + ...
%e A380057 where A( (exp(x) - 1)/cos(x) ) = exp(x).
%e A380057 SPECIFIC VALUES.
%e A380057 A(t) = 3/2 at t = 0.544117577471602714300218052469...
%e A380057 where t = (1/2)/real( (3/2)^i ) = (1/2)/cos(log(3/2)).
%e A380057 A(t) = 4/3 at t = 0.347619044845549508650339299345...
%e A380057 where t = (1/3)/cos(log(4/3)).
%e A380057 A(t) = 5/4 at t = 0.256355931938414599112107620645...
%e A380057 where t = (1/4)/cos(log(5/4)).
%e A380057 A(t) = 6/5 at t = 0.203370786475336228292002468204...
%e A380057 where t = (1/5)/cos(log(6/5)).
%e A380057 A(5/9) = 1.509166099041466745516450953830986339123819257989853587...
%e A380057 A(1/2) = 1.464101647089238181480821477030380839753029025795281424...
%e A380057 where A(1/2) = real(1 + (1/2)*A(1/2)^i) = 1 + (1/2)*cos(log( A(1/2) )).
%e A380057 A(1/3) = 1.320532220599397916493026637544832197337142648617761054...
%e A380057 where A(1/3) = 1 + (1/3)*cos(log( A(1/3) )).
%e A380057 A(1/4) = 1.244062312309475342914040842626056476267801020397804125...
%e A380057 where A(1/4) = 1 + (1/4)*cos(log( A(1/4) )).
%e A380057 A(1/5) = 1.196781755871960043253050643301603078428252760913031148...
%e A380057 A(1/6) = 1.164732612936027556202815971272406070730644075186904648...
%e A380057 A(1/10) = 1.099550027586687118480790742605384041459169163472092691...
%o A380057 (PARI) {a(n) = my(A = 1+O(x)); for(i=0, n, A = real( 1 + x*A^I )); n!*polcoef(A, n)}
%o A380057 for(n=0,30, print1(a(n),", "))
%Y A380057 Cf. A380053, A380055, A380058.
%K A380057 sign
%O A380057 0,4
%A A380057 _Paul D. Hanna_, Jan 24 2025