A201128 E.g.f. satisfies: A(x) = 1 + tan(x*A(x)).

1, 1, 2, 8, 56, 536, 6336, 89984, 1502080, 28816000, 624183040, 15068741632, 401279717376, 11686228873216, 369481496674304, 12604143513731072, 461450753988067328, 18047560103055097856, 750979001470906073088, 33127742823550018125824, 1544257684260700601712640

Offset

0,3

Comments

Radius of convergence of e.g.f. A(x) is r = 2*atan(Pi/2-1)/Pi = 0.3301951... where A(r) = Pi/2.

Radius of convergence of e.g.f. A(x) is not 2*atan(Pi/2-1)/Pi, but r = 0.39828426936099395926..., where r is the root of the equation cos(r + sqrt(r*(1-r))) = sqrt(r), and A(r) = 1 + sqrt((1-r)/r) = 2.22913364222177489... - Vaclav Kotesovec, Jan 11 2014

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Vaclav Kotesovec, Graph - asymptotics for 1000 terms

Formula

a(n) = [x^n/n!] (1 + tan(x))^(n+1)/(n+1).

E.g.f. satisfies:

(1) A(x/(1 + tan(x))) = 1 + tan(x).

(2) A(x) = (1/x)*Series_Reversion(x/(1 + tan(x))).

(3) A(x) = 1 + Sum_{n>=1} 8*x*A(x) / ((2*n-1)^2*Pi^2 - 4*x^2*A(x)^2).

(4) A(x) = 1 + x*A(x)/(1 - x^2*A(x)^2/(3 - x^2*A(x)^2/(5 - x^2*A(x)^2/(7 - x^2*A(x)^2/(9 - ...))))), a recursive continued fraction.

a(n) = (Sum_{k=0..n/2} binomial(n+1, 2*k+1)*(Sum_{j=0..2*k} binomial(j+n-2*k-1, n-2*k-1)*(j+n-2*k)!*2^(-j+2*k)*(-1)^(k+j)*Stirling2(n, j+n-2*k)))/(n+1), n > 0, a(0)=1. - Vladimir Kruchinin, Feb 03 2012

a(n) ~ n^(n-1) * sqrt((1 + 1/sqrt(1/r-1))/(2*r))/ (exp(n) * r^n), where r = 0.39828426936099395926... is the root of the equation cos(r + sqrt(r*(1-r))) = sqrt(r). - Vaclav Kotesovec, Jan 11 2014

Example

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 56*x^4/4! + 536*x^5/5! + ...
where A(x) = 1 + 8*x*A(x)/(Pi^2 - 4*x^2*A(x)^2) + 8*x*A(x)/(9*Pi^2 - 4*x^2*A(x)^2) + 8*x*A(x)/(25*Pi^2 - 4*x^2*A(x)^2) + 8*x*A(x)/(49*Pi^2 - 4*x^2*A(x)^2) + ...
The coefficients of x^n/n! in initial powers of G(x) = 1 + tan(x) begin:
G^1: [(1), 1, 0, 2, 0, 16, 0, 272, 0, 7936, ...];
G^2: [1,(2), 2, 4, 16, 32, 272, 544, 7936, ...];
G^3: [1, 3, (6), 12, 48, 168, 816, 4512, 23808, ...];
G^4: [1, 4, 12,(32), 120, 544, 2592, 15872, 96000, ...];
G^5: [1, 5, 20, 70, (280), 1400, 7520, 46720, 321280, ...];
G^6: [1, 6, 30, 132, 600, (3216), 19200, 125952, 925440, ...];
G^7: [1, 7, 42, 224, 1176, 6832, (44352), 312704, 2424576, ...];
G^8: [1, 8, 56, 352, 2128, 13568, 94976, (719872), 5907328, ...];
G^9: [1, 9, 72, 522, 3600, 25344, 191232, 1552752, (13518720), ...]; ...
where coefficients in parenthesis form initial terms of this sequence:
[1/1, 2/2, 6/3, 32/4, 280/5, 3216/6, 44352/7, 719872/8, 13518720/9, ...].

Mathematica

CoefficientList[1/x*InverseSeries[Series[x/(1 + Tan[x]), {x, 0, 21}], x], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 11 2014 *)
Table[Sum[Binomial[n+1, 2*k+1]*Sum[Binomial[j+n-2*k-1, n-2*k-1]*(j+n-2*k)!*2^(-j+2*k)*(-1)^(k+j)*StirlingS2[n, j+n-2*k]/(n+1), {j, 0, 2*k}], {k, 0, n/2}], {n, 0, 20}] (* Vaclav Kotesovec after Vladimir Kruchinin, Jan 11 2014 *)

Prog

(PARI) {a(n)=n!*polcoeff(1/x*serreverse(x/(1+tan(x+x^2*O(x^n)))), n)}
(Maxima) a(n):=if n=0 then 1 else sum(binomial(n+1, 2*k+1)*sum(binomial(j+n-2*k-1, n-2*k-1)*(j+n-2*k)!*2^(-j+2*k)*(-1)^(k+j)*stirling2(n, j+n-2*k), j, 0, 2*k), k, 0, (n)/2)/(n+1); /* Vladimir Kruchinin, Feb 03 2012 */

Crossrefs

Cf. A009006.

Sequence in context: A308501 A316142 A109618 * A277498 A097691 A181939

Adjacent sequences: A201125 A201126 A201127 * A201129 A201130 A201131

Keyword

nonn

Author

Paul D. Hanna, Dec 01 2011

Status

approved