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%I #54 Sep 08 2022 08:45:09
%S 1,1,11,131,1661,22101,305151,4335711,63009881,932449961,14004694451,
%T 212944033051,3271618296661,50711564152381,792088104593511,
%U 12454801769554551,196991734871121201,3131967533789345361,50026642742943415131,802406215117502069811
%N a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 10^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 10^C(n+1,2). - _Philippe Deléham_, Oct 29 2007
%C a(n) = upper left term in M^n, M = the production matrix:
%C 1, 1;
%C 10, 10, 10;
%C 1, 1, 1, 1;
%C 10, 10, 10, 10, 10;
%C 1, 1, 1, 1, 1, 1;
%C ...
%C - _Gary W. Adamson_, Jul 08 2011
%C Shifts left when INVERT transform applied ten times. - _Benedict W. J. Irwin_, Feb 07 2016
%C For fixed m > 0, if g.f. = (1+m*x-sqrt(m^2*x^2-(2*m+4)*x+1))/((2*m+2)*x) then a(n,m) ~ (m + 2 + 2*sqrt(m+1))^(n + 1/2) / (2*sqrt(Pi) * (m+1)^(3/4) * n^(3/2)). - _Vaclav Kotesovec_, Mar 19 2018
%H Vincenzo Librandi, <a href="/A082148/b082148.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+9*x-sqrt(81*x^2-22*x+1))/(20*x).
%F a(n) = Sum_{k=0..n} A088617(n, k)*10^k*(-9)^(n-k). - _Philippe Deléham_, Jan 21 2004
%F a(n) = (11*(2n-1)*a(n-1) - 81*(n-2)*a(n-2)) / (n+1) for n>=2, a(0)=a(1)=1. - _Philippe Deléham_, Aug 19 2005
%F a(n) ~ sqrt(20+11*sqrt(10))*(11+2*sqrt(10))^n/(20*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 14 2012
%F G.f.: 1/(1 - x/(1 - 10*x/(1 - x/(1 - 10*x/(1 - x/(1 - ...)))))), a continued fraction. - _Ilya Gutkovskiy_, Aug 10 2017
%F a(n) = hypergeom([1 - n, -n], [2], 10). - _Peter Luschny_, Mar 19 2018
%p A082148_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]+10*add(a[j]*a[w-j-1],j=1..w-1) od;
%p convert(a, list) end: A082148_list(17); # _Peter Luschny_, May 19 2011
%t Table[SeriesCoefficient[(1+9*x-Sqrt[81*x^2-22*x+1])/(20*x),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 14 2012 *)
%t a[n_] := Sum[10^k*1/n*Binomial[n, k]*Binomial[n, k + 1], {k, 0, n}];
%t a[0] = 1; Array[a, 20, 0] (* _Robert G. Wilson v_, Feb 10 2018 *)
%t a[n_] := Hypergeometric2F1[1 - n, -n, 2, 10];
%t Table[a[n], {n, 0, 18}] (* _Peter Luschny_, Mar 19 2018 *)
%o (PARI) a(n)=if(n<1,1,sum(k=0,n,10^k/n*binomial(n,k)*binomial(n,k+1)))
%o (Magma) I:=[1,11]; [1] cat [n le 2 select I[n] else (11*(2*n-1)*Self(n-1) - 81*(n-2)*Self(n-2))/(n+1): n in [1..30]]; // _G. C. Greubel_, Feb 10 2018
%Y Cf. A001003, A007564, A059231.
%K nonn
%O 0,3
%A _Benoit Cloitre_, May 10 2003
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