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%I #53 Jul 02 2023 20:49:00
%S 1,0,1,0,9,0,225,0,11025,0,893025,0,108056025,0,18261468225,0,
%T 4108830350625,0,1187451971330625,0,428670161650355625,0,
%U 189043541287806830625,0,100004033341249813400625,0,62502520838281133375390625,0,45564337691106946230659765625,0
%N Expansion of e.g.f. arcsin(x).
%C A001818 interspersed with zeros. - _Joerg Arndt_, Aug 31 2013
%C 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).
%C 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
%D L. Comtet and M. Fiolet, Sur les dérivées successives d'une fonction implicite. C. R. Acad. Sci. Paris Ser. A 278 (1974), 249-251.
%H Steven Finch, <a href="https://arxiv.org/abs/2111.14487">Rounds, Color, Parity, Squares</a>, arXiv:2111.14487 [math.CO], 2021.
%H Michael Wallner, <a href="https://arxiv.org/abs/1706.07163">A bijection of plane increasing trees with relaxed binary trees of right height at most one</a>, arXiv:1706.07163 [math.CO], 2017, p. 12.
%F E.g.f.: arcsin(x).
%F 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]
%F E.g.f of a(n + 1), n >= 0, is 1/sqrt(1 - x^2). - _N. Sato_, Aug 29 2013
%F If n is odd, a(n) ~ 2*n^(n-1) / exp(n). - _Vaclav Kotesovec_, Oct 05 2013
%F 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
%F a(n) = (n-1)! - A087137(n-1). - _Anton Zakharov_, Oct 18 2016
%e 1 is in the sequence because, for k=1, f'(x) = 1/sqrt(1-x^2), and f'(0) = 1.
%e G.f. = x + x^3 + 9*x^5 + 225*x^7 + 11025*x^9 + 893025*x^11 + ...
%p 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):
%t a[ n_] := If[ n < 1, 0, If[ EvenQ[n], 0, (n - 2)!!^2]]; (* _Michael Somos_, Oct 07 2013 *)
%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ ArcSin[x], {x, 0, n}]]; (* _Michael Somos_, Oct 07 2013 *)
%o (PARI) Vec( serlaplace( sqrt( 1/(1-x^2) + O(x^55) ) ) )
%o (PARI) {a(n) = if( n<2, n==1, (n-2)^2 * a(n-2))}; /* _Michael Somos_, Oct 07 2013 */
%o (PARI) a(n) = if( n<0, 0, n! * polcoeff( asin(x + x * O(x^n)), n)); /* _Michael Somos_, Oct 07 2013 */
%Y Alternate terms are A001818. - _N. Sato_, May 13 2010
%Y Cf. A087137.
%K nonn
%O 1,5
%A _Michel Lagneau_, May 03 2010
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