OFFSET
0,5
REFERENCES
Dwight, Tables of Integrals ..., Eq. 552.5, page 133.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..100
G. Guillotte and L. Carlitz, Problem H-216 and solution, Fib. Quarter. p. 90, Vol 13, 1, Feb. 1975.
R. Kelisky, The numbers generated by exp(arctan x), Duke Math. J., 26 (1959), 569-581.
H. P. Robinson and N. J. A. Sloane, Correspondence, 1971-1972
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
FORMULA
E.g.f.: exp(arctan(x)).
a(n) = n!*sum(if oddp(m+n) then 0 else (-1)^((3*n+m)/2)/(2^m*m!)*sum(2^i*binomial(n-1,i-1)*m!/i!*stirling1(i,m),i,m,n),m,1,n), n>0. - Vladimir Kruchinin, Aug 05 2010
E.g.f.: exp(arctan(x)) = 1 + 2x/(H(0)-x); H(k) = 4k + 2 + x^2*(4k^2 + 8k + 5)/H(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 15 2011
a(n+1) = a(n) - a(n-1) * A002378(n-2). - Reinhard Zumkeller, Feb 27 2012
E.g.f.: -2i*(B((1+ix)/2; (2-i)/2, (2+i)/2) - B(1/2; (2-i)/2, (2+i)/2)), for a(0)=0, a(1)=a(2)=a(3)=1, B(x;a,b) is the incomplete Beta function. - G. C. Greubel, May 01 2015
a(n) = i^n*n!*Sum_{r+s=n} (-1)^s*binomial(-i/2, r)*binomial(i/2,s) where i is the imaginary unit. See the Fib. Quart. link. - Michel Marcus, Jan 22 2017
MATHEMATICA
RecurrenceTable[{a[0]==1, a[1]==1, a[n]==a[n-1]-(n-1)(n-2)a[n-2]}, a[n], {n, 30}] (* Harvey P. Dale, May 02 2011 *)
CoefficientList[Series[E^(ArcTan[x]), {x, 0, 20}], x]*Range[0, 20]! (* Vaclav Kotesovec, Nov 06 2014 *)
PROG
(Maxima) a(n):=n!*sum(if oddp(m+n) then 0 else (-1)^((3*n+m)/2)/(2^m*m!)*sum(2^i*binomial(n-1, i-1)*m!/i!*stirling1(i, m), i, m, n), m, 1, n); /* Vladimir Kruchinin, Aug 05 2010 */
(Haskell)
a002019 n = a002019_list !! n
a002019_list = 1 : 1 : zipWith (-)
(tail a002019_list) (zipWith (*) a002019_list a002378_list)
-- Reinhard Zumkeller, Feb 27 2012
(Magma) I:=[1, 1]; [1] cat [ n le 2 select I[n] else Self(n-1)-(n^2-3*n+2)*Self(n-2): n in [1..35]]; // Vincenzo Librandi, May 02 2015
CROSSREFS
KEYWORD
sign,nice,easy
AUTHOR
EXTENSIONS
More terms from Herman P. Robinson
STATUS
approved