OFFSET
0,3
COMMENTS
Hankel transform of a(n) is A143750. Hankel transform of a(n+1) is 10^C(n+1,2).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
G.f.: (1-9*x-sqrt(1-22*x+81*x^2))/2.
a(n) = Sum_{k=0..n-1} C(n+k-1,2*k)*A000108(k)*9^(n-k-1).
a(n+1) = Sum_{k=0..n} C(2*n-k,k)*A000108(n-k)*9^k.
a(n+1) = 0^n + (1/(n+0^n)) * Sum_{k=0..n} C(n,k)*C(n,k-1)*10^k.
a(n+1) = Sum_{k, 0<=k<=n} 10^k*A090181(n,k). - Philippe Deléham, Oct 14 2008
a(n) = 9 * a(n-1) + Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
Recurrence: n*a(n) = 11*(2*n-3)*a(n-1) - 81*(n-3)*a(n-2) . - Vaclav Kotesovec, Oct 20 2012
a(n) ~ sqrt(11*sqrt(10)-20)*(11+2*sqrt(10))^n/(18*sqrt(Pi)*n^(3/2)) . - Vaclav Kotesovec, Oct 20 2012
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
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
EXAMPLE
G.f. = x + 10*x^2 + 110*x^3 + 1310*x^4 + 16610*x^5 + 221010*x^6 + 3051510*x^7 + ...
MATHEMATICA
CoefficientList[Series[(1-9*x-Sqrt[1-22*x+81*x^2])/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
PROG
(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 */
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Aug 30 2008
STATUS
approved