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%I #19 May 24 2013 11:25:22
%S 3,-4,12,-40,140,-504,1848,-6864,25740,-97240,369512,-1410864,5408312,
%T -20801200,80233200,-310235040,1202160780,-4667212440,18150270600,
%U -70690527600,275693057640,-1076515748880,4208197927440,-16466861455200
%N Z-sequence for the Riordan triangle A111125;
%C For the notion Z-sequence for a Riordan triangle (lower triangular matrix) R(n,m) see a W.Lang link under A006232, with references. The Z-sequence appears in the recurrence for any entry R(n,0), n >= 1: R(n,0) = sum(Z(m)*R(n-1,m), m=0..n-1).
%C The A-sequence for the Riordan triangle A111125 is (-1)^n*A115141(n).
%F O.g.f.: (1 - (2 - c(-x))/(1 + 4*x))/(1 - c(-x)) = ((3 + 4*x) + 4*x*c(-x))/(1 + 4*x), with c(x) the o.g.f. of A000108 (Catalan).
%F a(0) = 3, a(n) = ((-1)^n)*4*binomial(2*n-1,n), n >= 1, (from adding the two pieces of the second o.g.f. version).
%F |a(n)| = A100320(n), n >0. - _R. J. Mathar_, Apr 22 2013
%F G.f.: 1 + G(0), where G(k)= 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) - (k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 24 2013
%Y Cf.: A111125, A115141.
%K sign,easy
%O 0,1
%A _Wolfdieter Lang_, Oct 18 2012
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