A380053
E.g.f. (exp(x) - 1)/cos(x).
Original entry on oeis.org
1, 1, 4, 7, 36, 91, 624, 2087, 18256, 76231, 814144, 4078867, 51475776, 300870571, 4381112064, 29265244847, 482962852096, 3629392540111, 66942218896384, 558956224522027, 11394877025289216, 104659828714136851, 2336793875186479104, 23414201065072302407, 568240131312188379136
Offset: 1
E.g.f.: A(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! + 76231*x^10/10! + ...
RELATED SERIES.
1/cos(x) = 1 + x^2/2! + 5*x^4/4! + 61*x^6/6! + 1385*x^8/8! + 50521*x^10/10! + 2702765*x^12/12! + 199360981*x^14/14! + ... + A000364(n)*x^(2*n)/(2*n)! + ...
Let F(x) be the series reversion of e.g.f. A(x) so that A(F(x)) = x, then
F(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 F(x) = log(1 + x*cos(F(x))).
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{a(n) = my(X = x + x*O(x^n)); n!*polcoef( (exp(X) - 1)/cos(X), n)}
for(n=1,25,print1(a(n),", "))
A380055
E.g.f. satisfies A(x) = log( 1 + x*cos(A(x)) ).
Original entry on oeis.org
1, -1, -1, 18, -86, -210, 8840, -80080, -266220, 19991520, -274725100, -1006434000, 123657316600, -2328145274000, -8148732243600, 1621702497792000, -39454300872662000, -113331522571488000, 38748502249144766000, -1172806114215446464000, -2126467491228525900000, 1525200888587905488960000
Offset: 1
E.g.f.: 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! + 19991520*x^10/10! - 274725100*x^11/11! + ...
where exp(A(x)) = 1 + x*cos(A(x)).
RELATED SERIES.
exp(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! + ... + A380057(n)*x^n/n! + ...
cos(A(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! + ...
(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! + ... + A380053(n)*x^n/n! + ...
where A( (exp(x) - 1)/cos(x) ) = x.
SPECIAL VALUES.
A(t) = Pi/10 at t = 0.388102877484829654642911664361938906648...
where t = (exp(Pi/10) - 1) * sqrt(2/5) * sqrt(5 - sqrt(5)).
A(t) = Pi/12 at t = 0.309822824437268302125213263289623021697...
where t = (exp(Pi/12) - 1) * sqrt(2) * (sqrt(3) - 1).
A(t) = Pi/16 at t = 0.221202550489384066264418133743972067830...
where t = (exp(Pi/16) - 1) * 2/sqrt(2 + sqrt(2 + sqrt(2))).
SPECIFIC VALUES.
A(1/3) = 0.27803485275982692914601139870101331536340692581609...
where A(1/3) = log( 1 + (1/3)*cos(A(1/3)) ).
A(1/4) = 0.21838208334344020266585373465948828480095045415275...
where A(1/4) = log( 1 + (1/4)*cos(A(1/4)) ).
A(1/5) = 0.17963608403094093235073066420053332711004361034830...
where A(1/5) = log( 1 + (1/5)*cos(A(1/5)) ).
A(1/6) = 0.15249154388154448186532142316863060968215944337447...
where A(1/6) = log( 1 + (1/6)*cos(A(1/6)) ).
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{a(n) = my(X = x + x*O(x^n)); n!*polcoef( serreverse( (exp(X) - 1)/cos(X) ), n)}
for(n=1,25,print1(a(n),", "))
A380556
E.g.f. A(x) satisfies A(x) = real( 1 + x*A(x)^(2*i) ), where i^2 = -1.
Original entry on oeis.org
1, 1, 0, -12, 48, 820, -14160, -69160, 5900160, -44796960, -3089865600, 88646729600, 1412786918400, -135956951062400, 1023512450688000, 203887248898944000, -7307555382586368000, -252816835499795840000, 26110132266648748032000, -95216226972043640320000, -80962066973581160140800000
Offset: 0
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)).
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CoefficientList[Exp[InverseSeries[Series[ (Exp[x] - 1)/Cos[2*x] ,{x,0,20}],x]],x]Range[0,20]! (* Stefano Spezia, Jan 29 2025 *)
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{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),", "))
Showing 1-3 of 3 results.