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 A201561 E.g.f. satisfies: A(x) = x + tan( A(x) )^2 with A(0)=0. 1
 1, 2, 12, 136, 2160, 43952, 1092672, 32102656, 1088252160, 41809041152, 1795201638912, 85196352787456, 4428299422310400, 250187205957220352, 15265712890413023232, 1000468694343925006336, 70089639485229413498880, 5227049493330884279140352, 413441163603081566484037632 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Table of n, a(n) for n=1..19. FORMULA E.g.f.: Series_Reversion( x - tan(x)^2 ). E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) tan(x)^(2*n)/n!. E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (tan(x)^(2*n)/x)/n! ). a(n) ~ t*sqrt(((1+arccos(t))*t^2-1)/(6-4*t^2)) * n^(n-1) / (exp(n) * (1+arccos(t)-1/t^2)^n), where t = sqrt(((6*(9+sqrt(129)))^(1/3) - 2*6^(2/3)/(9+sqrt(129))^(1/3))/3) = 0.920710376904467468... is the root of the equation 4-4*t^2 = t^6. - Vaclav Kotesovec, Jan 12 2014 EXAMPLE E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 136*x^4/4! + 2160*x^5/5! +... where A(x - tan(x)^2) = x. Related expansions: A(x) = x + tan(x)^2 + d/dx tan(x)^4/2! + d^2/dx^2 tan(x)^6/3! + d^3/dx^3 tan(x)^8/4! +... log(A(x)/x) = tan(x)^2/x + d/dx (tan(x)^4/x)/2! + d^2/dx^2 (tan(x)^6/x)/3! + d^3/dx^3 (tan(x)^8/x)/4! +... tan(A(x)) = x + 2*x^2/2! + 14*x^3/3! + 160*x^4/4! + 2536*x^5/5! + 51632*x^6/6! +... tan(A(x))^2 = 2*x^2/2! + 12*x^3/3! + 136*x^4/4! + 2160*x^5/5! +... tan(x) = x + 2*x^3/3! + 16*x^5/5! + 272*x^7/7! + 7936*x^9/9! +... tan(x)^2 = 2*x^2/2! + 16*x^4/4! + 272*x^6/6! + 7936*x^8/8! +... MATHEMATICA Rest[CoefficientList[InverseSeries[Series[x - Tan[x]^2, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 12 2014 *) PROG (PARI) a(n, m=1)=n!*polcoeff(serreverse(x-tan(x+x*O(x^n))^2), n) (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, tan(x+x*O(x^n))^(2*m)/m!)); n!*polcoeff(A, n)} (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, tan(x+x*O(x^n))^(2*m)/x/m!)+x*O(x^n))); n!*polcoeff(A, n)} for(n=1, 25, print1(a(n), ", ")) CROSSREFS Cf. A205886. Sequence in context: A289998 A180353 A208873 * A205886 A108996 A117513 Adjacent sequences: A201558 A201559 A201560 * A201562 A201563 A201564 KEYWORD nonn AUTHOR Paul D. Hanna, Dec 02 2011 STATUS approved

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Last modified June 9 19:10 EDT 2023. Contains 363183 sequences. (Running on oeis4.)