A002019 a(n) = a(n-1) - (n-1)(n-2)a(n-2).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• nice
• easy

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.

Crossrefs

Bisections are A102058 and A102059.

Cf. A006228.

Row sums of signed triangle A049218.

Cf. A000246.

Programs

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

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 *)

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 */

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

Extensions

More terms from Herman P. Robinson