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A182122 Expansion of exp( arcsinh( 2*x ) ).

%I A182122 #33 Jun 20 2025 08:11:40
%S A182122 1,2,2,0,-2,0,4,0,-10,0,28,0,-84,0,264,0,-858,0,2860,0,-9724,0,33592,
%T A182122 0,-117572,0,416024,0,-1485800,0,5348880,0,-19389690,0,70715340,0,
%U A182122 -259289580,0,955277400,0,-3534526380,0,13128240840,0,-48932534040,0,182965127280
%N A182122 Expansion of exp( arcsinh( 2*x ) ).
%H A182122 G. C. Greubel, <a href="/A182122/b182122.txt">Table of n, a(n) for n = 0..1000</a>
%F A182122 G.f.: 2*x + sqrt( 1 + 4*x^2 ) = 1 / (1 - 2*x / (1 + x / (1 - x / (1 + x / ... )))).
%F A182122 The g.f. A(x) satisfies: A(x) = sqrt(1 + 4*x * A(x)).
%F A182122 a(n) = (-1)^n * A104624(n). Convolution inverse of A104624.
%F A182122 Conjecture : n*(n+1)*a(n) + (n+2)*(n-1)*a(n-1) +4*(n+1)*(n-3)*a(n-2) +4*(n+2)*(n-4)*a(n-3) = 0.- _R. J. Mathar_, Jul 24 2012
%F A182122 a(n) = 2*hypergeom([-n+1,2-n],[2],-1). - _Peter Luschny_, Sep 23 2014
%F A182122 0 = a(n)*(+16*a(n+2) + 10*a(n+4)) + a(n+2)*(-2*a(n+2) + a(n+4)) if n>=0. - _Michael Somos_, Jan 10 2017
%F A182122 a(n+4) = 2 * a(n+2) * (a(n+2) - 8*a(n)) / (a(n+2) + 10*a(n)) if n>=0 is even. - _Michael Somos_, Jan 10 2017
%F A182122 G.f. A(x) satisfies A(x) = 1/A(-x). - _Seiichi Manyama_, Jun 20 2025
%e A182122 G.f. = 1 + 2*x + 2*x^2 - 2*x^4 + 4*x^6 - 10*x^8 + 28*x^10 - 84*x^12 + ...
%p A182122 s := proc(n) option remember; `if`(n<2, n+1, -4*(n-2)*s(n-2)/(n+1)) end: A127846 := n -> `if`(n<2,n+1,s(n-1)); seq(A127846(n), n=0..47); # _Peter Luschny_, Sep 23 2014
%t A182122 CoefficientList[Series[Exp[ArcSinh[2x]],{x,0,50}],x] (* _Harvey P. Dale_, Aug 18 2012 *)
%t A182122 Table[2 HypergeometricPFQ[{-n+1,2-n},{2},-1],{n,0,46}] (* _Peter Luschny_, Sep 23 2014 *)
%o A182122 (PARI) {a(n) = if( n<2, (n>=0) + (n>0), n = n-2; if( n%2, 0, (-1)^(n/2) * 4 * binomial( n, n/2) / (n + 2)))};
%o A182122 (PARI) {a(n) = if( n<0, 0, polcoeff( sqrt( 1 + 4*x^2 + x*O(x^n) ) + 2*x, n ) )};
%o A182122 (PARI) {a(n) = my(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = sqrt( 1 + 4*x * A)); polcoeff( A, n))};
%o A182122 (Sage)
%o A182122 def A182122(n):
%o A182122     if n < 2: return n+1
%o A182122     if n % 2 == 1: return 0
%o A182122     return (-1)^(n/2-1)*binomial(n,n/2)/(n-1)
%o A182122 [A182122(n) for n in range(47)] # _Peter Luschny_, Sep 23 2014
%o A182122 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Exp(Argsinh(2*x)))); // _G. C. Greubel_, Aug 12 2018
%Y A182122 Cf. A104624.
%Y A182122 Cf. A000984, A048990, A078531, A224884, A245112, A245113.
%K A182122 sign
%O A182122 0,2
%A A182122 _Michael Somos_, Apr 13 2012