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A006229
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Expansion of exp( tan x ).
(Formerly M2822)
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13
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1, 1, 1, 3, 9, 37, 177, 959, 6097, 41641, 325249, 2693691, 24807321, 241586893, 2558036145, 28607094455, 342232522657, 4315903789009, 57569080467073, 807258131578995, 11879658510739497, 183184249105857781, 2948163649552594737, 49548882107764546223
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OFFSET
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0,4
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 259, Sum_{k} T(n,k).
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 150.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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E.g.f.: exp(tan(x)).
a(n) = sum(if oddp(n+k) then 0 else (-1)^((n+k)/2)*sum(j!/k!*stirling2(n,j)*2^(n-j)*(-1)^(n+j-k)*binomial(j-1,k-1),j,k,n),k,1,n), n>0. - Vladimir Kruchinin, Aug 05 2010
E.g.f.: 1 + tan(x)/T(0), where T(k) = 4*k+1 - tan(x)/(2 + tan(x)/(4*k+3 - tan(x)/(2 + tan(x)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 03 2013
a(n) = sum(i=0..(n-1)/2, binomial(n-1,2*i)*z(i)*a(n-2*i-1)), a(0)=1, where z(n) is tangent (or "zag") numbers (A000182). - Vladimir Kruchinin, Mar 04 2015
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MATHEMATICA
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With[{nn=30}, CoefficientList[Series[Exp[Tan[x]], {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Oct 04 2011 *)
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PROG
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(Maxima)
a(n):=sum(if oddp(n+k) then 0 else (-1)^((n+k)/2)*sum(j!/k!*stirling2(n, j)*2^(n-j)*(-1)^(n+j-k)*binomial(j-1, k-1), j, k, n), k, 1, n); /* Vladimir Kruchinin, Aug 05 2010 */
(Julia)
len <= 0 && return BigInt[]
T = zeros(BigInt, len, len); T[1, 1] = 1
S = Array(BigInt, len); S[1] = 1
for n in 2:len
T[n, n] = 1
for k in 2:n-1 T[n, k] = T[n-1, k-1] + k*(k-1)*T[n-1, k+1] end
S[n] = sum(T[n, k] for k in 2:n)
end
S end
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Feb 08 2001
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STATUS
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approved
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