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A082147 a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 8^k*N(n,k) where N(n,k) = (1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

%I A082147 #57 Sep 08 2022 08:45:09
%S A082147 1,1,9,89,945,10577,123129,1476841,18130401,226739489,2878666857,
%T A082147 37006326777,480750990993,6301611631473,83240669582937,
%U A082147 1106980509493641,14808497812637121,199138509770855489,2690461489090104009
%N A082147 a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 8^k*N(n,k) where N(n,k) = (1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).
%C A082147 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 A082147 The Hankel transform of this sequence is 8^C(n+1,2). - _Philippe Deléham_, Oct 29 2007
%C A082147 Shifts left when INVERT transform applied eight times. - _Benedict W. J. Irwin_, Feb 07 2016
%H A082147 Vincenzo Librandi, <a href="/A082147/b082147.txt">Table of n, a(n) for n = 0..200</a>
%H A082147 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 A082147 G.f.: (1 + 7*x - sqrt(49*x^2-18*x+1))/(16*x).
%F A082147 a(n) = Sum_{k=0..n} A088617(n, k)*8^k*(-7)^(n-k). - _Philippe Deléham_, Jan 21 2004
%F A082147 a(n) = (9(2n-1)a(n-1) - 49(n-2)a(n-2)) / (n+1) for n >= 2, a(0) = a(1) = 1. - _Philippe Deléham_, Aug 19 2005
%F A082147 a(n) = upper left term in M^n, M = the production matrix:
%F A082147   1, 1
%F A082147   8, 8, 8
%F A082147   1, 1, 1, 1
%F A082147   8, 8, 8, 8, 8
%F A082147   1, 1, 1, 1, 1, 1
%F A082147   ...
%F A082147 - _Gary W. Adamson_, Jul 08 2011
%F A082147 a(n) ~ sqrt(16+18*sqrt(2))*(9+4*sqrt(2))^n/(16*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 14 2012
%F A082147 G.f.: 1/(1 - x/(1 - 8*x/(1 - x/(1 - 8*x/(1 - x/(1 - ...)))))), a continued fraction. - _Ilya Gutkovskiy_, Apr 21 2017
%F A082147 a(n) = hypergeom([1 - n, -n], [2], 8). - _Peter Luschny_, Mar 19 2018
%p A082147 A082147_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
%p A082147 for w from 1 to n do a[w] := a[w-1]+8*add(a[j]*a[w-j-1],j=1..w-1) od;
%p A082147 convert(a, list) end: A082147_list(18); # _Peter Luschny_, May 19 2011
%t A082147 Table[SeriesCoefficient[(1+7*x-Sqrt[49*x^2-18*x+1])/(16*x),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 14 2012 *)
%t A082147 f[n_] := Sum[ 8^k*Binomial[n, k]*Binomial[n, k + 1]/n, {k, 0, n}]; f[0] = 1; Array[f, 21, 0] (* _Robert G. Wilson v_, Feb 24 2018 *)
%t A082147 a[n_] := Hypergeometric2F1[1 - n, -n, 2, 8];
%t A082147 Table[a[n], {n, 0, 18}] (* _Peter Luschny_, Mar 19 2018 *)
%o A082147 (PARI) a(n)=if(n<1,1,sum(k=0,n,8^k/n*binomial(n,k)*binomial(n,k+1)))
%o A082147 (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1+7*x-Sqrt(49*x^2-18*x+1))/(16*x))) // _G. C. Greubel_, Feb 05 2018
%o A082147 (GAP) a:=n->Sum([0..n],k->8^k*(1/n)*Binomial(n,k)*Binomial(n,k+1));;
%o A082147 Concatenation([1],List([1..18],n->a(n))); # _Muniru A Asiru_, Feb 10 2018
%Y A082147 Cf. A001003, A007564, A059231.
%K A082147 nonn
%O A082147 0,3
%A A082147 _Benoit Cloitre_, May 10 2003