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A003720
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Expansion of e.g.f. tan(tan(tan(x))).
(Formerly M4301)
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2
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1, 6, 168, 10672, 1198080, 208521728, 51874413568, 17449541107712, 7622674735988736, 4193561606973095936, 2836052065377836597248, 2312174256451088534208512, 2236165580390456719589769216, 2530976708469616321520834969600, 3314110602212685014002135203840000
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OFFSET
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0,2
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REFERENCES
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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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a(n) = b(2*n+1) where b(n) = sum(m=1..n, (((-1)^(m-1)+1)*(sum(j=1..m, j! *2^(m-j-1)*(-1)^((m+1)/2+j)*S2(m,j)))*sum(k=m..n,(((-1)^(k-m)+1)*(sum(j=m..k, C(j-1,m-1)*j!*2^(k-j-1)*S2(k,j)*(-1)^((m+k)/2+j)))*((-1)^(n-k)+1)* sum(j=k,n, C(j-1,k-1)*j!*2^(n-j-1)* (-1)^((n+k)/2+j)* S2(n,j)))/k!))/m!). - Vladimir Kruchinin, Apr 23 2011
a(n) ~ 8*(2*n+1)!/((4+Pi^2) * (1+arctan(Pi/2)^2) * (arctan(arctan(Pi/2)))^(2*n+2)). - Vaclav Kotesovec, Feb 16 2015
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MATHEMATICA
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Rest@ Union[ Range[0, 25]! CoefficientList[ Series[Tan@ Tan@ Tan@ x, {x, 0, 25}], x]] (* Robert G. Wilson v *)
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PROG
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(Maxima) a(n):=b(2*n+1); b(n):=sum((((-1)^(m-1)+1)*(sum(j!*2^(m-j-1)* (-1)^((m+1)/2+j) *stirling2(m, j), j, 1, m))*sum((((-1)^(k-m)+1)*(sum(binomial(j-1, m-1)* j!*2^(k-j-1)*stirling2(k, j)*(-1)^((m+k)/2+j), j, m, k))*((-1)^(n-k)+1)* sum(binomial(j-1, k-1)*j!*2^(n-j-1)* (-1)^((n+k)/2+j)* stirling2(n, j) , j, k, n))/k!, k, m, n))/m!, m, 1, n); [Vladimir Kruchinin, Apr 23 2011]
(PARI) x='x+O('x^66); /* that many terms */
serlaplace(tan(tan(tan(x)))) /* show terms */ /* Joerg Arndt, Apr 26 2011 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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