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A104624 Expansion of exp( arcsinh( -2*x ) ) in powers of x.

%I A104624 #57 Jan 07 2024 10:53:31
%S A104624 1,-2,2,0,-2,0,4,0,-10,0,28,0,-84,0,264,0,-858,0,2860,0,-9724,0,33592,
%T A104624 0,-117572,0,416024,0,-1485800,0,5348880,0,-19389690,0,70715340,0,
%U A104624 -259289580,0,955277400,0,-3534526380,0,13128240840,0,-48932534040,0,182965127280,0,-686119227300,0
%N A104624 Expansion of exp( arcsinh( -2*x ) ) in powers of x.
%C A104624 First column in inverse of A054335.
%C A104624 With offset 1 the coefficient sequence of series reversion of A000984 (binomial(2n,n)) also with offset 1. - _Michael Somos_, Jan 14 2011
%H A104624 G. C. Greubel, <a href="/A104624/b104624.txt">Table of n, a(n) for n = 0..1000</a>
%H A104624 Rigoberto Flórez, Leandro Junes, and José L. Ramírez, <a href="https://doi.org/10.1016/j.disc.2019.06.018">Enumerating several aspects of non-decreasing Dyck paths</a>, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3091.
%H A104624 Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F A104624 G.f.: sqrt( 1 + 4*x^2 ) - 2*x = exp( asinh( -2*x ) ). - _Michael Somos_, Jan 14 2011
%F A104624 The positive sequence 1,2,2,0,2,... has g.f. 2(1+x)-sqrt(1-4x^2). - _Paul Barry_, Oct 10 2007
%F A104624 From _Vladimir Kruchinin_, Jan 16 2011: (Start)
%F A104624 The o.g.f. A(x) satisfies A(x)=x*sqrt(1-4*A(x)),
%F A104624 a(n) = 1/(n*(n+1)) * sum(j=0...n+1, j * 2^(j) * binomial(2*n-j-1,n-1) * binomial(n+1,j) * (-1)^(n-j)). (End)
%F A104624 Conjecture: n*a(n) + (n-1)*a(n-1) + 4*(n-3)*a(n-2) + 4*(n-4)*a(n-3) = 0. - _R. J. Mathar_, Nov 15 2012
%F A104624 If n is even, a(n) ~ (-1)^(1+n/2) * 2^(n+1) * n^(n-1) / exp(n). - _Vaclav Kotesovec_, Oct 23 2013
%F A104624 G.f.: 2*S(0) -1-2*x-4*x^2, where S(k) = 2*k+1 + x^2*(2*k+3)/(1 + x^2*(2*k+1)/S(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Dec 23 2013
%F A104624 a(n) = (-1)^n*2*hypergeom([-n+1, 2-n], [2], -1). - _Peter Luschny_, Sep 23 2014
%e A104624 G.f. = 1 - 2*x + 2*x^2 - 2*x^4 + 4*x^6 - 10*x^8 + 28*x^10 - 84*x^12 + 264*x^14 + ...
%p A104624 s := proc(n) option remember; `if`(n<2, n+1, 4*(n-2)*s(n-2)/(n+1)) end: A104624 := n -> `if`(n<2, (-1)^n*(n+1), (-1)^(n/2-1)*s(n-1)); seq(A104624(n), n=0..47); # _Peter Luschny_, Sep 23 2014
%t A104624 CoefficientList[ Series[ Exp[ ArcSinh[ -2x]], {x, 0, 49}], x]
%t A104624 Table[(-1)^n 2 HypergeometricPFQ[{-n+1, 2-n}, {2}, -1], {n, 0, 46}] (* _Peter Luschny_, Sep 23 2014 *)
%o A104624 (PARI) {a(n) = if( n<0, 0, polcoeff( sqrt( 1 + 4*x^2 + x*O(x^n) ) - 2*x, n ) )}; /* _Michael Somos_, Jan 14 2011 */
%o A104624 (Sage)
%o A104624 def A104624(n):
%o A104624     if n < 2: return (-1)^n*(n+1)
%o A104624     if n % 2 == 1: return 0
%o A104624     return (-1)^(n/2+1)*binomial(n,n/2)/(n-1)
%o A104624 [A104624(n) for n in range(47)] # _Peter Luschny_, Sep 23 2014
%o A104624 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Exp(Argsinh(-2*x)))); // _G. C. Greubel_, Aug 12 2018
%K A104624 easy,sign
%O A104624 0,2
%A A104624 _Paul Barry_, Mar 17 2005