A177145 Expansion of e.g.f. arcsin(x).

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

Offset: 1

Author

Michel Lagneau, May 03 2010

Keywords

• nonn
• easy

Comments

A001818 interspersed with zeros. - Joerg Arndt, Aug 31 2013

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

Examples

			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 + ...
		

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

• Muhammad Adam Dombrowski and Gregory Dresden, Areas Between Cosines, arXiv:2404.17694 [math.CO], 2024. See p. 11.
• Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
• Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017, p. 12.

Crossrefs

Alternate terms are A001818. - N. Sato, May 13 2010

Cf. A087137.

Programs

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 *)

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 */

Formula

E.g.f.: arcsin(x).

G.f.: Q(0)*x^2/(1+x) + x/(1+x), where Q(k) = 1 + (2*k + 1)^2 * x * (1 + x * Q(k+1)); - Sergei N. Gladkovskii, May 10 2013 [Edited by Michael Somos, Oct 07 2013]

E.g.f of a(n+1), n >= 0, is 1/sqrt(1 - x^2). - N. Sato, Aug 29 2013

If n is odd, a(n) ~ 2*n^(n-1) / exp(n). - Vaclav Kotesovec, Oct 05 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

a(n) = (n-1)! - A087137(n-1). - Anton Zakharov, Oct 18 2016

From Peter Bala, Aug 09 2024: (Start)

a(2*n+1) = (2*n - 1)!!^2 = A001147(n)^2.

a(n) = (n - 2)^2 * a(n-2) with a(1) = 1 and a(2) = 0. (End)