A143749 - OEIS

%I #32 Oct 02 2024 01:58:40

%S 0,1,10,110,1310,16610,221010,3051510,43357110,630098810,9324499610,

%T 140046944510,2129440330510,32716182966610,507115641523810,

%U 7920881045935110,124548017695545510,1969917348711212010

%N Series reversion of x * (1 - x) / (1 + 9*x).

%C Hankel transform of a(n) is A143750. Hankel transform of a(n+1) is 10^C(n+1,2).

%H Vincenzo Librandi, <a href="/A143749/b143749.txt">Table of n, a(n) for n = 0..200</a>

%F G.f.: (1-9*x-sqrt(1-22*x+81*x^2))/2.

%F a(n) = Sum_{k=0..n-1} C(n+k-1,2*k)*A000108(k)*9^(n-k-1).

%F a(n+1) = Sum_{k=0..n} C(2*n-k,k)*A000108(n-k)*9^k.

%F a(n+1) = 0^n + (1/(n+0^n)) * Sum_{k=0..n} C(n,k)*C(n,k-1)*10^k.

%F a(n+1) = Sum_{k, 0<=k<=n} 10^k*A090181(n,k). - _Philippe Deléham_, Oct 14 2008

%F a(n) = 9 * a(n-1) + Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - _Michael Somos_, Jul 23 2011

%F Recurrence: n*a(n) = 11*(2*n-3)*a(n-1) - 81*(n-3)*a(n-2) . - _Vaclav Kotesovec_, Oct 20 2012

%F a(n) ~ sqrt(11*sqrt(10)-20)*(11+2*sqrt(10))^n/(18*sqrt(Pi)*n^(3/2)) . - _Vaclav Kotesovec_, Oct 20 2012

%F 0 = a(n)*(6561*a(n+1) - 4455*a(n+2) + 324*a(n+3)) + a(n+1)*(891*a(n+1) + 322*a(n+2) - 55*a(n+3)) + a(n+2)*(11*a(n+2) + a(n+3)) if n>0. - _Michael Somos_, Jan 23 2014

%F G.f.: x/(1 - 9*x - x/(1 - 9*x - x/(1 - 9*x - x/(1 - 9*x - x/(1 - ...))))), a continued fraction. - _Ilya Gutkovskiy_, Apr 07 2018

%e G.f. = x + 10*x^2 + 110*x^3 + 1310*x^4 + 16610*x^5 + 221010*x^6 + 3051510*x^7 + ...

%t CoefficientList[Series[(1-9*x-Sqrt[1-22*x+81*x^2])/2, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 20 2012 *)

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

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-9*x-Sqrt(81*x^2-22*x+1))/(2*x))); // _G. C. Greubel_, Sep 16 2018

%Y Cf. A000108, A090181, A143750.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Aug 30 2008