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A012101 Expansion of tan(arcsin(sinh(x))) (odd powers only). 2
1, 4, 76, 3424, 277456, 35345344, 6504742336, 1632531979264, 535821754153216, 222769351470429184, 114411762387714436096, 71132353206363509039104, 52648938670226334981246976 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..219

FORMULA

a(n) = Sum_{m=0..n} ( binomial(2*m,m)*2^(-4*m)*( Sum_{i=0,..,(2*m+1)/2} (2*i-2*m-1)^(2*n+1)*binomial(2*m+1,i)*(-1)^(i+1) ) ). - Vladimir Kruchinin, Jun 15 2011

E.g.f.: sinh(x) / sqrt(1 - sinh(x)^2). - Vaclav Kotesovec, Feb 06 2015

a(n) ~ (2*n+1)! / (sqrt(Pi*n) * 2^(1/4) * (log(1+sqrt(2)))^(2*n+3/2)). - Vaclav Kotesovec, Feb 06 2015

EXAMPLE

tan(arcsin(sinh(x))) = x+4/3!*x^3+76/5!*x^5+3424/7!*x^7+277456/9!*x^9...

MATHEMATICA

nn = 20; Table[(CoefficientList[Series[Sinh[x]/Sqrt[1 - Sinh[x]^2], {x, 0, 2*nn+1}], x] * Range[0, 2*nn+1]!)[[n]], {n, 2, 2*nn, 2}] (* Vaclav Kotesovec, Feb 06 2015 *)

Table[Sum[Binomial[2*m, m]*2^(-4*m)*Sum[(2*i - 2*m - 1)^(2*n + 1)*

Binomial[2*m + 1, i]*(-1)^(i + 1), {i, 0, (2*m + 1)/2}], {m, 0, n}], {n, 0, 50}] (* G. C. Greubel, Feb 15 2017 *)

PROG

(Maxima)

a(n):=sum(binomial(2*m, m)*2^(-4*m)*sum((2*i-2*m-1)^(2*n+1)*binomial(2*m+1, i)*(-1)^(i+1), i, 0, (2*m+1)/2), m, 0, n); /* Vladimir Kruchinin, Jun 15 2011 */

CROSSREFS

Cf. A012571.

Sequence in context: A012020 A012041 A024258 * A012080 A012047 A012010

Adjacent sequences:  A012098 A012099 A012100 * A012102 A012103 A012104

KEYWORD

nonn

AUTHOR

Patrick Demichel (patrick.demichel(AT)hp.com)

EXTENSIONS

Typo in second formula corrected (following a suggestion of Sergei N. Gladkovskii) by Vaclav Kotesovec, Apr 17 2017

STATUS

approved

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Last modified August 17 19:38 EDT 2022. Contains 356189 sequences. (Running on oeis4.)