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A006229
Expansion of e.g.f. exp( tan x ).
(Formerly M2822)
(Formerly M2822)
13
1, 1, 1, 3, 9, 37, 177, 959, 6097, 41641, 325249, 2693691, 24807321, 241586893, 2558036145, 28607094455, 342232522657, 4315903789009, 57569080467073, 807258131578995, 11879658510739497, 183184249105857781, 2948163649552594737, 49548882107764546223
OFFSET
0,4
REFERENCES
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).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..480 (terms 0..100 from T. D. Noe)
J. Shallit, Letter to N. J. A. Sloane, May 1975
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
FORMULA
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+1)*a(n-2*i-1), a(0)=1, where z(n) is tangent (or "zag") numbers (A000182). - Vladimir Kruchinin, Mar 04 2015 [corrected by Jason Yuen, Dec 29 2024]
MATHEMATICA
With[{nn=30}, CoefficientList[Series[Exp[Tan[x]], {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Oct 04 2011 *)
PROG
(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)
function A006229_list(len::Int)
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
println(A006229_list(24)) # Peter Luschny, Apr 27 2017
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Feb 08 2001
STATUS
approved