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

0,4

Links

Table of n, a(n) for n=0..23.

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

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

Example

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

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 *)

Prog

(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 */

Crossrefs

Cf. A109033.

Sequence in context: A277871 A082307 A099782 * A110276 A026883 A349730

Adjacent sequences: A109031 A109032 A109033 * A109035 A109036 A109037

Keyword

nonn

Author

Emeric Deutsch, Jun 16 2005

Status

approved