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%I
%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.
%D R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, 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>
%F G.f.: (1+2x-sqrt(1+4x^2))/(2x)
%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). [From Paul Barry, Jan 15 2009]
%F a(n) = sum{k=0..n, A090181(n,k)*(-1)^k}. [From Philippe DELEHAM, Feb 02 2009]
%F a(n) = (1/n)*sum((-2)^i*binomial(n, i)*binomial(2*n-i-2, n-1), i=0..n-1). [From Vladimir Kruchinin, Dec 26 2010]
%F With offset 1, then 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 Conjecture: (n+1)*a(n) +n*a(n-1) +4*(n-2)*a(n-2) +4*(n-3)*a(n-3)=0. - R. J. Mathar, Nov 15 2011
%e 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}] (* From Jean-François Alcover, Nov 14 2011, after Vladimir Kruchinin *)
%o {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 */
%Y Cf. A097331, A090192.
%K easy,sign
%O 0,6
%A _Paul Barry_, Apr 11 2005
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