|
|
A177145
|
|
Expansion of e.g.f. arcsin(x).
|
|
10
|
|
|
1, 0, 1, 0, 9, 0, 225, 0, 11025, 0, 893025, 0, 108056025, 0, 18261468225, 0, 4108830350625, 0, 1187451971330625, 0, 428670161650355625, 0, 189043541287806830625, 0, 100004033341249813400625, 0, 62502520838281133375390625, 0, 45564337691106946230659765625, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,5
|
|
COMMENTS
|
a(n) is the number of permutations of n-1 where all cycles have even length. For example, a(5)=9 and the permutations of 4 elements with only even cycles are (1,2)(3,4); (1,3)(2,4); (1,4)(2,3); (1,2,3,4); (1,2,4,3); (1,3,2,4); (1,3,4,2); (1,4,2,3); (1,4,3,2).
a(n) is the number of permutations on n - 1 elements where there are no cycles of even length and an even number of cycles of odd length. - N. Sato, Aug 29 2013
|
|
REFERENCES
|
L. Comtet and M. Fiolet, Sur les dérivées successives d'une fonction implicite. C. R. Acad. Sci. Paris Ser. A 278 (1974), 249-251.
|
|
LINKS
|
|
|
FORMULA
|
E.g.f.: arcsin(x).
E.g.f of a(n + 1), n >= 0, is 1/sqrt(1 - x^2). - N. Sato, Aug 29 2013
E.g.f.: arcsin(x) = x + x^3/(T(0)-x^2), where T(k) = 4*k^2*(1+x^2) + 2*k*(5+2*x^2) +6 + x^2 - 2*x^2*(k+1)*(2*k+3)^3/T(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 13 2013
|
|
EXAMPLE
|
1 is in the sequence because, for k=1, f'(x) = 1/sqrt(1-x^2), and f'(0) = 1.
G.f. = x + x^3 + 9*x^5 + 225*x^7 + 11025*x^9 + 893025*x^11 + ...
|
|
MAPLE
|
n0:= 30: T:=array(1..n0+1): f:=x->arcsin(x):for n from 1 to n0 do:T[n]:=(D(f)(0)):f:=D(f):od: print(T):
|
|
MATHEMATICA
|
a[ n_] := If[ n < 1, 0, If[ EvenQ[n], 0, (n - 2)!!^2]]; (* Michael Somos, Oct 07 2013 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ ArcSin[x], {x, 0, n}]]; (* Michael Somos, Oct 07 2013 *)
|
|
PROG
|
(PARI) Vec( serlaplace( sqrt( 1/(1-x^2) + O(x^55) ) ) )
(PARI) {a(n) = if( n<2, n==1, (n-2)^2 * a(n-2))}; /* Michael Somos, Oct 07 2013 */
(PARI) a(n) = if( n<0, 0, n! * polcoeff( asin(x + x * O(x^n)), n)); /* Michael Somos, Oct 07 2013 */
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|