A002019 - OEIS

%I M4330 N1813 #67 Jun 01 2024 11:44:24

%S 1,1,1,-1,-7,5,145,-5,-6095,-5815,433025,956375,-46676375,-172917875,

%T 7108596625,38579649875,-1454225641375,-10713341611375,

%U 384836032842625,3663118565923375,-127950804666254375,-1519935859717136875

%N a(n) = a(n-1) - (n-1)(n-2)a(n-2).

%D Dwight, Tables of Integrals ..., Eq. 552.5, page 133.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A002019/b002019.txt">Table of n, a(n) for n = 0..100</a>

%H G. Guillotte and L. Carlitz, <a href="http://www.fq.math.ca/Scanned/13-1/advanced13-1.pdf">Problem H-216 and solution</a>, Fib. Quarter. p. 90, Vol 13, 1, Feb. 1975.

%H R. Kelisky, <a href="http://dx.doi.org/10.1215/S0012-7094-59-02654-7">The numbers generated by exp(arctan x)</a>, Duke Math. J., 26 (1959), 569-581.

%H H. P. Robinson and N. J. A. Sloane, <a href="/A002037/a002037.pdf">Correspondence, 1971-1972</a>

%H Kruchinin Vladimir Victorovich, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.

%F E.g.f.: exp(arctan(x)).

%F 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

%F 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

%F a(n+1) = a(n) - a(n-1) * A002378(n-2). - _Reinhard Zumkeller_, Feb 27 2012

%F 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

%F 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

%t 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 *)

%t CoefficientList[Series[E^(ArcTan[x]),{x,0,20}],x]*Range[0,20]! (* _Vaclav Kotesovec_, Nov 06 2014 *)

%o (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 */

%o (Haskell)

%o a002019 n = a002019_list !! n

%o a002019_list = 1 : 1 : zipWith (-)

%o    (tail a002019_list) (zipWith (*) a002019_list a002378_list)

%o -- _Reinhard Zumkeller_, Feb 27 2012

%o (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

%Y Bisections are A102058 and A102059.

%Y Cf. A006228.

%Y Row sums of signed triangle A049218.

%Y Cf. A000246.

%K sign,nice,easy

%O 0,5

%A _N. J. A. Sloane_

%E More terms from _Herman P. Robinson_