A109034 First differences of A109033.

1, 0, 1, 4, 16, 66, 280, 1216, 5384, 24224, 110464, 509480, 2372704, 11142656, 52709600, 250933120, 1201354240, 5780413760, 27937867520, 135574988800, 660314620160, 3226731934720, 15815752724480, 77735943378560

Offset: 0

Author

Emeric Deutsch, Jun 16 2005

Keywords

• nonn

Examples

			G.f. = 1 + x^2 + 4*x^3 + 16*x^4 + 66*x^5 + 280*x^6 + 1216*x^7 + 5384*x^8 + ...
		

Links

• Ian Le, Wilf classes of pairs of permutations of length 4, Electron. J. Combin., 12(1) (2005), R25, 26 pages.

Crossrefs

Cf. A109033.

Programs

Maple

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);

Mathematica

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 *)

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 */

Formula

G.f. A(x) = y satisfies 0 = 2*x*y^2 - y + (1-x)^2. - Michael Somos, Jan 05 2012

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

G.f.: 2 * (1 - x)^2 / (1 + sqrt(1 - 8*x + 16*x^2 - 8*x^3)). - Michael Somos, Jan 05 2012

G.f. = (1 - sqrt(1 - 8*x + 16*x^2 - 8*x^3))/(4*x).

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

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