A002019 a(n) = a(n-1) - (n-1)(n-2)a(n-2). (Formerly M4330 N1813)

1, 1, 1, -1, -7, 5, 145, -5, -6095, -5815, 433025, 956375, -46676375, -172917875, 7108596625, 38579649875, -1454225641375, -10713341611375, 384836032842625, 3663118565923375, -127950804666254375, -1519935859717136875

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

Bisections are A102058 and A102059.

Cf. A006228.

Row sums of signed triangle A049218.

Cf. A000246.

Sequence in context: A007553 A294474 A248277 * A012878 A300452 A302201

Adjacent sequences: A002016 A002017 A002018 * A002020 A002021 A002022

Keyword

sign,nice,easy

Author

N. J. A. Sloane

Extensions

More terms from Herman P. Robinson

Status

approved