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 A082181 a(0)=1; for n>=1, a(n) = sum(k=0..n, 9^k*N(n,k)), where N(n,k) =1/n*C(n,k)*C(n,k+1) are the Narayana numbers (A001263). 7
 1, 1, 10, 109, 1270, 15562, 198100, 2596645, 34825150, 475697854, 6595646860, 92590323058, 1313427716380, 18798095833012, 271118225915560, 3936516861402901, 57494017447915150, 844109420603623030 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS More generally, coefficients of (1+m*x-sqrt(m^2*x^2-(2*m+4)*x+1))/((2*m+2)*x) are given by: a(n) = sum(k=0..n, (m+1)^k*N(n,k)). The Hankel transform of this sequence is 9^C(n+1,2). - Philippe Deléham, Oct 29 2007 a(n) = upper left term in M^n, M = the production matrix: 1, 1 9, 9, 9 1, 1, 1, 1 9, 9, 9, 9, 9 1, 1, 1, 1, 1, 1 ... - Gary W. Adamson, Jul 08 2011 Shifts left when INVERT transform applied nine times. - Benedict W. J. Irwin, Feb 07 2016 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4. FORMULA G.f.: (1+8*x-sqrt(64*x^2-20*x+1))/(18*x). a(n) = Sum_{k=0..n} A088617(n, k)*9^k*(-8)^(n-k). - Philippe Deléham, Jan 21 2004 a(n) = (10(2n-1)a(n-1) - 64(n-2)a(n-2)) / (n+1) for n>=2, a(0)=a(1)=1. - Philippe Deléham, Aug 19 2005 a(n) ~ 2^(4*n+1)/(3*sqrt(3*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012 G.f.: 1/(1 - x/(1 - 9*x/(1 - x/(1 - 9*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Apr 21 2017 a(n) = hypergeom([1 - n, -n], [2], 9). - Peter Luschny, Mar 19 2018 MAPLE A082181_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1; for w from 1 to n do a[w] := a[w-1]+9*add(a[j]*a[w-j-1], j=1..w-1) od; convert(a, list) end: A082181_list(17); # Peter Luschny, May 19 2011 MATHEMATICA Table[SeriesCoefficient[(1+8*x-Sqrt[64*x^2-20*x+1])/(18*x), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *) a[n_] := Hypergeometric2F1[1 - n, -n, 2, 9]; Table[a[n], {n, 0, 18}] (* Peter Luschny, Mar 19 2018 *) PROG (PARI) a(n)=if(n<1, 1, sum(k=0, n, 9^k/n*binomial(n, k)*binomial(n, k+1))) CROSSREFS Cf. A001003, A007564, A059231. Sequence in context: A015591 A078922 A199760 * A190919 A095740 A075508 Adjacent sequences:  A082178 A082179 A082180 * A082182 A082183 A082184 KEYWORD nonn AUTHOR Benoit Cloitre, May 10 2003 EXTENSIONS Corrected by T. D. Noe, Oct 25 2006 STATUS approved

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Last modified August 6 10:01 EDT 2020. Contains 336245 sequences. (Running on oeis4.)