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A012150
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Expansion of e.g.f. exp(tan(arcsin(x))).
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1
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1, 1, 1, 4, 13, 76, 421, 3256, 25369, 245008, 2449801, 28441216, 346065061, 4700478784, 67243537453, 1047088053376, 17192488230961, 302112622479616, 5593309059948049, 109527844826856448, 2255588021494237501
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OFFSET
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0,4
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LINKS
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Table of n, a(n) for n=0..20.
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
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FORMULA
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From Vladimir Kruchinin, Feb 17 2011: (Start)
a(n) = n!*Sum_{k=1..n} A111959(n-1,k-1)*2^(k-n)/k!.
a(n) = n!*Sum_{k=1..n} (1+(-1)^(n-k))*C((n-2)/2,(n-k)/2)/(2*k!), n>0.
E.g.f.: exp(x/sqrt(1-x^2)). (End)
E.g.f.: S(x) = exp(x/sqrt(1-x^2)) = 1 + 2*(x/sqrt(1-x^2))/(G(0) - x/sqrt(1-x^2)), G(k) = 8*k + 2 + (x^2)/((1-x^2)*(8*k+6) + x^2/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Dec 16 2011
a(n) = (3*n^2 - 12*n + 13)*a(n-2) - 3*(n-4)*(n-3)^2*(n-2)*a(n-4) + (n-6)*(n-5)*(n-4)^2*(n-3)*(n-2)*a(n-6). - Vaclav Kotesovec, Nov 08 2013
a(n) ~ n^(n-1/3) * exp(3/2*n^(1/3)-n) / sqrt(3) * (1 - 19/(36*n^(1/3)) + 553/(2592*n^(2/3))). - Vaclav Kotesovec, Nov 08 2013
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EXAMPLE
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exp(tan(arcsin(x))) = 1+x+1/2!*x^2+4/3!*x^3+13/4!*x^4+76/5!*x^5...
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MAPLE
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A012150 := proc(n) if n = 0 then 1; else add( (1+(-1)^(n-k)) *binomial((n-2)/2, (n-k)/2)/(2*k!), k=1..n) ; %*n! ; end if; end proc: # R. J. Mathar, Mar 20 2011
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MATHEMATICA
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Range[0, 20]! CoefficientList[Series[Exp[Tan[ArcSin[x]]], {x, 0, 20}], x] (* Or *)
f[n_] := n! Sum[(1 + (-1)^(n - k)) Binomial[(n - 2)/2, (n - k)/2]/2/k!, {k, n}]; f[0] = 1; Array[f, 21, 0] (* Robert G. Wilson v, Feb 19 2011 *)
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PROG
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(PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(tan(asin(x))))) \\ Michel Marcus, Oct 30 2022
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CROSSREFS
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Cf. A111959.
Sequence in context: A171756 A235385 A144055 * A012261 A012075 A197942
Adjacent sequences: A012147 A012148 A012149 * A012151 A012152 A012153
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KEYWORD
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nonn
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AUTHOR
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Patrick Demichel (patrick.demichel(AT)hp.com)
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EXTENSIONS
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Name edited by Michel Marcus, Oct 30 2022
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STATUS
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approved
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