|
| |
|
|
A002019
|
|
a(n)=a(n-1)-(n-1)(n-2)a(n-2).
(Formerly M4330 N1813)
|
|
8
|
|
|
|
1, 1, 1, -1, -7, 5, 145, -5, -6095, -5815, 433025, 956375, -46676375, -172917875, 7108596625, 38579649875, -1454225641375, -10713341611375, 384836032842625, 3663118565923375, -127950804666254375, -1519935859717136875
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
REFERENCES
|
Dwight, Tables of Integrals ..., Eq. 552.5, page 133.
R. Kelisky, The numbers generated by exp(arctan x), Duke Math. J., 26 (1959), 569-581.
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
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565
|
|
|
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. [From 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]
|
|
|
MAPLE
|
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); (for Maxima) [From Vladimir Kruchinin, Aug 05 2010]
|
|
|
MATHEMATICA
|
RecurrenceTable[{a[0]==1, a[1]==1, a[n]==a[n-1]-(n-1)(n-2)a[n-2]}, a[n], {n, 30}] (* From Harvey P. Dale, May 02 2011 *)
|
|
|
PROG
|
(Haskell)
a002019 n = a002019_list !! n
a002019_list = 1 : 1 : zipWith (-)
(tail a002019_list) (zipWith (*) a002019_list a002378_list)
-- Reinhard Zumkeller, Feb 27 2012
|
|
|
CROSSREFS
|
Bisections are A102058 and A102059.
Cf. A006228.
Row sums of signed triangle A049218.
Cf. A000246.
Sequence in context: A005692 A080798 A007553 * A012878 A003299 A198677
Adjacent sequences: A002016 A002017 A002018 * A002020 A002021 A002022
|
|
|
KEYWORD
|
sign,nice,easy
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
EXTENSIONS
|
More terms from H. P. Robinson.
|
|
|
STATUS
|
approved
|
| |
|
|
