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A012494
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Expansion of arctan(sin(x)).
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1
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1, -3, 45, -1743, 125625, -14554683, 2473184805, -579439207623, 179018972217585, -70518070842040563, 34495620120141463965, -20515677772241956573503, 14578232896601243652363945
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| arctan(cos(x)*tan(x))=x-3/3!*x^3+45/5!*x^5-1743/7!*x^7+125625/9!*x^9...
Absolute values are coefficients in expansion of atanh(asinh(x)).
arctanh(sinh(x)) = x + 3*x^3/3! + 45*x^5/5! + 1743*x^7/7! +...
arccot(sin(x)) = Pi/2 - x + 3*x^3/3! - 45*x^5/5! + 1743*x^7/7! -...
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FORMULA
| a(n):=n!*sum(k=1..ceiling(n/2), (1+(-1)^(n-2*k+1))*2^(1-2*k)*sum(i=0..(2*k-1)/2, (-1)^((n+1)/2-i)*binomial(2*k-1,i)*(2*i-2*k+1)^n/n!)/(2*k-1)), n>0. [From Vladimir Kruchinin, Feb 25 2011]
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MATHEMATICA
| Drop[ Range[0, 25]! CoefficientList[ Series[ ArcTan[ Sin[x]], {x, 0, 25}], x], {1, 25, 2}] (* Or *)
f[n_] := n!Sum[(1 + (-1)^(n - 2k + 1))2^(1 - 2k)Sum[(-1)^((n + 1)/2 - j)Binomial[2k - 1, j]((2j - 2k + 1)^n/n!)/(2k - 1), {j, 0, (2k - 1)/2}], {k, Ceiling[n/2]}]; Table[ f[n], {n, 1, 25, 2}] (* RGWv *)
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PROG
| (Maxima):
a(n):=n!*sum((1+(-1)^(n-2*k+1))*2^(1-2*k)*sum((-1)^((n+1)/2-i)*binomial(2*k-1, i)*(2*i-2*k+1)^n/n!, i, 0, (2*k-1)/2)/(2*k-1), k, 1, ceiling((n)/2)); [From Vladimir Kruchinin, Feb 25 2011]
a(n):=sum(sum((2*i-2*k-1)^(2*n+1)*binomial(2*k+1, i)*(-1)^(n-i+1), i, 0, k)/(4^k*(2*k+1)), k, 0, n); [From Vladimir Kruchinin, Feb 04 2012]
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CROSSREFS
| Bisection of A003704, A013208.
Cf. A101923.
Sequence in context: A144950 A144951 A079484 * A012780 A072503 A154242
Adjacent sequences: A012491 A012492 A012493 * A012495 A012496 A012497
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KEYWORD
| sign,changed
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AUTHOR
| Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
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