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%I #70 Apr 28 2024 11:34:24
%S 1,-1,0,1,0,-2,0,5,0,-14,0,42,0,-132,0,429,0,-1430,0,4862,0,-16796,0,
%T 58786,0,-208012,0,742900,0,-2674440,0,9694845,0,-35357670,0,
%U 129644790,0,-477638700,0,1767263190,0
%N Expansion of 1-x*c(-x^2) where c(x) is the g.f. of A000108.
%C Row sums of A105522. Row sums of inverse of A105438.
%C First column of number triangle A106180.
%H Vincenzo Librandi, <a href="/A105523/b105523.txt">Table of n, a(n) for n = 0..1000</a>
%H R. J. Martin and M. J. Kearney, <a href="http://dx.doi.org/10.1007/s00010-010-0051-0">An exactly solvable self-convolutive recurrence</a>, Aequat. Math., 80 (2010), 291-318. see p. 313.
%H R. J. Martin and M. J. Kearney, <a href="http://arXiv.org/abs/1103.4936">An exactly solvable self-convolutive recurrence</a>, arXiv:1103.4936 [math.CO], 2011.
%F G.f.: (1 + 2*x - sqrt(1+4*x^2))/(2*x).
%F a(n) = 0^n + sin(Pi*(n-2)/2)(C((n-1)/2)(1-(-1)^n)/2).
%F G.f.: 1/(1+x/(1-x/(1+x/(1-x/(1+x/(1-x.... (continued fraction). - _Paul Barry_, Jan 15 2009
%F a(n) = Sum{k = 0..n} A090181(n,k)*(-1)^k. - _Philippe Deléham_, Feb 02 2009
%F a(n) = (1/n)*sum_{i = 0..n-1} (-2)^i*binomial(n, i)*binomial(2*n-i-2, n-1). - _Vladimir Kruchinin_, Dec 26 2010
%F With offset 1, a(n) = -2 * a(n-1) + Sum_{k=1..n-1} a(k) * a(n-k), for n>1. - _Michael Somos_, Jul 25 2011
%F D-finite with recurrence: (n+3)*a(n+2) = -4*n*a(n), a(0)=1, a(1)=-1. - _Fung Lam_, Mar 18 2014
%F For nonzero terms, a(n) ~ (-1)^((n+1)/2)/sqrt(2*Pi)*2^(n+1)/(n+1)^(3/2). - _Fung Lam_, Mar 17 2014
%F a(n) = -(sqrt(Pi)*2^(n-1))/(Gamma(1-n/2)*Gamma((n+3)/2)) for n odd. - _Peter Luschny_, Oct 31 2014
%F From _Peter Bala_, Apr 20 2024: (Start)
%F a(n) = Sum_{k = 0..n} (-2)^(n-k)*binomial(n + k, 2*k)*Catalan(k), where Catalan(k) = A000108(k).
%F a(n) = (-2)^n * hypergeom([-n, n+1], [2], 1/2).
%F O.g.f.: A(x) = 1/x * series reversion of x*(1 - x)/(1 - 2*x). Cf. A152681. (End)
%e G.f. = 1 - x + x^3 - 2*x^5 + 5*x^7 - 14*x^9 + 42*x^11 - 132*x^13 + 429*x^15 + ...
%p A105523_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
%p for w from 1 to n do a[w]:=-a[w-1]+(-1)^w*add(a[j]*a[w-j-1],j=1..w-1) od; convert(a,list)end: A105523_list(40); # _Peter Luschny_, May 19 2011
%t a[n_?EvenQ] := 0; a[n_?OddQ] := 4^n*Gamma[n/2] / (Gamma[-n/2]*(n+1)!); a[0] = 1; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Nov 14 2011, after _Vladimir Kruchinin_ *)
%t CoefficientList[Series[(1 + 2 x - Sqrt[1 + 4 x^2])/(2 x), {x, 0, 50}], x] (* _Vincenzo Librandi_, Nov 01 2014 *)
%t a[ n_] := SeriesCoefficient[ (1 + 2 x - Sqrt[ 1 + 4 x^2]) / (2 x), {x, 0, n}]; (* _Michael Somos_, Jun 17 2015 *)
%t a[ n_] := If[ n < 1, Boole[n == 0], a[n] = -2 a[n - 1] + Sum[ a[j] a[n - j - 1], {j, 0, n - 1}]]; (* _Michael Somos_, Jun 17 2015 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, n++; A = vector(n); A[1] = 1; for( k=2, n, A[k] = -2 * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* _Michael Somos_, Jul 24 2011 */
%o (Sage)
%o def A105523(n):
%o if is_even(n): return 0 if n>0 else 1
%o return -(sqrt(pi)*2^(n-1))/(gamma(1-n/2)*gamma((n+3)/2))
%o [A105523(n) for n in (0..29)] # _Peter Luschny_, Oct 31 2014
%o (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1 + 2*x - Sqrt(1+4*x^2))/(2*x))); // _G. C. Greubel_, Sep 16 2018
%Y Cf. A000108, A097331, A090192, A090181, A105522, A105438.
%K easy,sign
%O 0,6
%A _Paul Barry_, Apr 11 2005
%E Typo in definition corrected by _Robert Israel_, Oct 31 2014
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