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A109034 First differences of A109033.

%I A109034 #31 Jul 24 2022 11:16:27
%S A109034 1,0,1,4,16,66,280,1216,5384,24224,110464,509480,2372704,11142656,
%T A109034 52709600,250933120,1201354240,5780413760,27937867520,135574988800,
%U A109034 660314620160,3226731934720,15815752724480,77735943378560
%N A109034 First differences of A109033.
%H A109034 Ian Le, <a href="http://www.combinatorics.org/Volume_12/Abstracts/v12i1r25.html">Wilf classes of pairs of permutations of length 4</a>, Electron. J. Combin., 12(1) (2005), R25, 26 pages.
%F A109034 G.f. A(x) = y satisfies 0 = 2*x*y^2 - y + (1-x)^2. - _Michael Somos_, Jan 05 2012
%F A109034 Given g.f. A(x), then B(x) = (A(x) - 1) / x satisfies B(-B(-x)) = x and B(x) - x = 4 * (B(x) * x) + 2 * (B(x) * x)^2. - _Michael Somos_, Jan 05 2012
%F A109034 G.f.: 2 * (1 - x)^2 / (1 + sqrt(1 - 8*x + 16*x^2 - 8*x^3)). - _Michael Somos_, Jan 05 2012
%F A109034 G.f. = (1 - sqrt(1 - 8*x + 16*x^2 - 8*x^3))/(4*x).
%F A109034 a(n) ~ 5^(1/4) * 2^(n-2) * phi^(2*n + 1/2) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 10 2021
%F A109034 D-finite with recurrence (n+1)*a(n) +4*(-2*n+1)*a(n-1) +16*(n-2)*a(n-2) +4*(-2*n+7)*a(n-3)=0. - _R. J. Mathar_, Jul 24 2022
%e A109034 G.f. = 1 + x^2 + 4*x^3 + 16*x^4 + 66*x^5 + 280*x^6 + 1216*x^7 + 5384*x^8 + ...
%p A109034 G:=(1-sqrt(1-8*x+16*x^2-8*x^3))/4/x: Gser:=series(G,x=0,30): 1,seq(coeff(Gser,x^n),n=1..27);
%t A109034 Join[{1},Differences[CoefficientList[Series[(1-Sqrt[1-8x+16x^2-8x^3])/ (4x(1-x)),{x,0,30}],x]]] (* _Harvey P. Dale_, Jul 06 2011 *)
%o A109034 (PARI) {a(n) = if( n<0, 0, polcoeff( 2 * (1 - x)^2 / (1 + sqrt(1 - 8*x + 16*x^2 - 8*x^3 + x * O(x^n))), n))} /* _Michael Somos_, Jan 05 2012 */
%Y A109034 Cf. A109033.
%K A109034 nonn
%O A109034 0,4
%A A109034 _Emeric Deutsch_, Jun 16 2005