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%I #51 May 24 2022 04:59:28
%S 1,1,10,109,1270,15562,198100,2596645,34825150,475697854,6595646860,
%T 92590323058,1313427716380,18798095833012,271118225915560,
%U 3936516861402901,57494017447915150,844109420603623030
%N 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).
%C 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).
%C The Hankel transform of this sequence is 9^C(n+1,2). - _Philippe Deléham_, Oct 29 2007
%C From _Gary W. Adamson_, Jul 08 2011: (Start)
%C a(n) = upper left term in M^n, M = the production matrix:
%C 1, 1
%C 9, 9, 9
%C 1, 1, 1, 1
%C 9, 9, 9, 9, 9
%C 1, 1, 1, 1, 1, 1
%C ... (End)
%C Shifts left when INVERT transform applied nine times. - _Benedict W. J. Irwin_, Feb 07 2016
%H Vincenzo Librandi, <a href="/A082181/b082181.txt">Table of n, a(n) for n = 0..200</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%F G.f.: (1+8*x-sqrt(64*x^2-20*x+1))/(18*x).
%F a(n) = Sum_{k=0..n} A088617(n, k)*9^k*(-8)^(n-k). - _Philippe Deléham_, Jan 21 2004
%F a(n) = (10*(2*n-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
%F a(n) ~ 2^(4*n+1)/(3*sqrt(3*Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 14 2012
%F G.f.: 1/(1 - x/(1 - 9*x/(1 - x/(1 - 9*x/(1 - x/(1 - ...)))))), a continued fraction. - _Ilya Gutkovskiy_, Apr 21 2017
%F a(n) = hypergeom([1 - n, -n], [2], 9). - _Peter Luschny_, Mar 19 2018
%p A082181_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
%p 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;
%p convert(a, list) end: A082181_list(17); # _Peter Luschny_, May 19 2011
%t 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 *)
%t a[n_] := Hypergeometric2F1[1 - n, -n, 2, 9];
%t Table[a[n], {n, 0, 18}] (* _Peter Luschny_, Mar 19 2018 *)
%o (PARI) a(n)=if(n<1,1,sum(k=0,n,9^k/n*binomial(n,k)*binomial(n,k+1)))
%o (Magma) [(&+[Binomial(n,k)*Binomial(n-1,k)*9^k/(k+1): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, May 23 2022
%o (SageMath) [sum(binomial(n,k)*binomial(n-1,k)*9^k/(k+1) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, May 23 2022
%Y Cf. A001003, A001263, A007564, A059231, A088617.
%K nonn
%O 0,3
%A _Benoit Cloitre_, May 10 2003
%E Corrected by _T. D. Noe_, Oct 25 2006
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