A006229 Expansion of exp( tan x ). (Formerly M2822)

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)*a(n-2*i-1)), a(0)=1, where z(n) is tangent (or "zag") numbers (A000182). - Vladimir Kruchinin, Mar 04 2015

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

Row sums of A059419 and unsigned A111593.

Cf. A003711, A003717, A000182, A047691, A047692.

Sequence in context: A077365 A366433 A319119 * A321734 A008986 A105215

Adjacent sequences: A006226 A006227 A006228 * A006230 A006231 A006232

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Jeffrey Shallit

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Feb 08 2001

Status

approved