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

%I A177145 #61 Aug 19 2024 15:31:28
%S A177145 1,0,1,0,9,0,225,0,11025,0,893025,0,108056025,0,18261468225,0,
%T A177145 4108830350625,0,1187451971330625,0,428670161650355625,0,
%U A177145 189043541287806830625,0,100004033341249813400625,0,62502520838281133375390625,0,45564337691106946230659765625,0
%N A177145 Expansion of e.g.f. arcsin(x).
%C A177145 A001818 interspersed with zeros. - _Joerg Arndt_, Aug 31 2013
%C A177145 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 A177145 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 A177145 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 A177145 Muhammad Adam Dombrowski and Gregory Dresden, <a href="https://arxiv.org/abs/2404.17694">Areas Between Cosines</a>, arXiv:2404.17694 [math.CO], 2024. See p. 11.
%H A177145 Steven Finch, <a href="https://arxiv.org/abs/2111.14487">Rounds, Color, Parity, Squares</a>, arXiv:2111.14487 [math.CO], 2021.
%H A177145 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 A177145 E.g.f.: arcsin(x).
%F A177145 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 A177145 E.g.f of a(n+1), n >= 0, is 1/sqrt(1 - x^2). - _N. Sato_, Aug 29 2013
%F A177145 If n is odd, a(n) ~ 2*n^(n-1) / exp(n). - _Vaclav Kotesovec_, Oct 05 2013
%F A177145 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 A177145 a(n) = (n-1)! - A087137(n-1). - _Anton Zakharov_, Oct 18 2016
%F A177145 From _Peter Bala_, Aug 09 2024: (Start)
%F A177145 a(2*n+1) = (2*n - 1)!!^2 = A001147(n)^2.
%F A177145 a(n) = (n - 2)^2 * a(n-2) with a(1) = 1 and a(2) = 0. (End)
%e A177145 1 is in the sequence because, for k=1, f'(x) = 1/sqrt(1-x^2), and f'(0) = 1.
%e A177145 G.f. = x + x^3 + 9*x^5 + 225*x^7 + 11025*x^9 + 893025*x^11 + ...
%p A177145 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 A177145 a[ n_] := If[ n < 1, 0, If[ EvenQ[n], 0, (n - 2)!!^2]]; (* _Michael Somos_, Oct 07 2013 *)
%t A177145 a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ ArcSin[x], {x, 0, n}]]; (* _Michael Somos_, Oct 07 2013 *)
%o A177145 (PARI) Vec( serlaplace( sqrt( 1/(1-x^2) + O(x^55) ) ) )
%o A177145 (PARI) {a(n) = if( n<2, n==1, (n-2)^2 * a(n-2))}; /* _Michael Somos_, Oct 07 2013 */
%o A177145 (PARI) a(n) = if( n<0, 0, n! * polcoeff( asin(x + x * O(x^n)), n)); /* _Michael Somos_, Oct 07 2013 */
%Y A177145 Alternate terms are A001818. - _N. Sato_, May 13 2010
%Y A177145 Cf. A087137.
%K A177145 nonn,easy
%O A177145 1,5
%A A177145 _Michel Lagneau_, May 03 2010