A131927 Expansion of series reversion of x * (1 - 9*x) / (1 - x).

0, 1, 8, 136, 2888, 68680, 1749896, 46707976, 1289214152, 36496595656, 1053849164552, 30918300671368, 919029058099784, 27617782977715528, 837674888992142984, 25610757376777402888, 788450850824647610312

Offset: 0

Author

Philippe Deléham, Oct 29 2007

Keywords

• nonn

Comments

The Hankel transform of this sequence is 72^C(n+1,2).

Examples

			G.f. = x + 8*x^2 + 136*x^3 + 2888*x^4 + 68680*x^5 + 1749896*x^6 + 46707976*x^7 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..650

Programs

Maple

G:=(1+x-sqrt(1-34*x+x^2))*1/18: Gser:=series(G,x=0,21): seq(coeff(Gser,x,n), n =0..17) # Emeric Deutsch, Nov 19 2007

Mathematica

CoefficientList[InverseSeries[Series[x*(1-9*x)/(1-x),{x,0,20}],x],x] (* Vaclav Kotesovec, Aug 20 2013 *)
a[ n_] := SeriesCoefficient[ (1 + x - Sqrt[ 1 - 34 x + x^2]) / 18, {x, 0, n}]; (* Michael Somos, Aug 30 2014 *)
a[ n_] := If[ n < 1, 0, SeriesCoefficient[ InverseSeries[ Series[ x (1 - 9 x) / (1 - x), {x, 0, n}]], {x, 0, n}]]; (* Michael Somos, Aug 30 2014 *)

PARI

{a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1;for( k=2, n, A[k] = - A[k-1] + 9 * sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 23 2011 */

Formula

a(n) = Sum_{k=0..n} A086810(n,k)*8^k.

G.f.: (1+x-sqrt(1-34*x+x^2))/18. - Emeric Deutsch, Nov 19 2007

a(n) = - a(n-1) + 9 * Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011

Recurrence: n*a(n) = 17*(2*n-3)*a(n-1) - (n-3)*a(n-2). - Vaclav Kotesovec, Aug 20 2013

a(n) ~ sqrt(102*sqrt(2)-144) * (17+12*sqrt(2))^n/(18*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 20 2013

0 = +a(n)*(+a(n+1) - 85*a(n+2) + 4*a(n+3)) + a(n+1)*(+17*a(n+1) + 1154*a(n+2) - 85*a(n+3)) + a(n+2)*(+17*a(n+2) + a(n+3)) for all n>0. - Michael Somos, Aug 30 2014

G.f.: x/(1 - 8*x/(1 - 9*x/(1 - 8*x/(1 - 9*x/(1 - 8*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Apr 21 2017

Extensions

More terms from Emeric Deutsch, Nov 19 2007