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A082173 a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 11^k*N(n,k) where N(n,k) = (1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

%I A082173 #51 Jan 21 2024 09:27:23
%S A082173 1,1,12,155,2124,30482,453432,6936799,108507180,1727970542,
%T A082173 27924685416,456820603086,7550600079672,125905525750500,
%U A082173 2115511349837040,35782547891727495,608787760350045420,10411451736723707990
%N A082173 a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 11^k*N(n,k) where N(n,k) = (1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).
%C A082173 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 A082173 The Hankel transform of this sequence is 11^C(n+1,2). - _Philippe Deléham_, Oct 29 2007
%C A082173 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 A082173 Vincenzo Librandi, <a href="/A082173/b082173.txt">Table of n, a(n) for n = 0..200</a>
%F A082173 G.f.: (1+10*x-sqrt(100*x^2-24*x+1))/(22*x).
%F A082173 a(n) = Sum_{k=0..n} A088617(n, k)*11^k*(-10)^(n-k). - _Philippe Deléham_, Jan 21 2004
%F A082173 a(n) = (12*(2n-1)*a(n-1) - 100*(n-2)*a(n-2)) / (n+1) for n >= 2, a(0) = a(1) = 1. - _Philippe Deléham_, Aug 19 2005
%F A082173 From _Gary W. Adamson_, Jul 08 2011: (Start)
%F A082173 a(n) = upper left term in M^n, M = the production matrix:
%F A082173    1,  1
%F A082173   11, 11, 11
%F A082173    1,  1,  1,  1
%F A082173   11, 11, 11, 11, 11
%F A082173    1,  1,  1,  1,  1,  1
%F A082173   ... (End)
%F A082173 a(n) ~ sqrt(22+12*sqrt(11))*(12+2*sqrt(11))^n/(22*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 14 2012
%F A082173 G.f.: 1/(1 - x/(1 - 11*x/(1 - x/(1 - 11*x/(1 - x/(1 - ...)))))), a continued fraction. - _Ilya Gutkovskiy_, Aug 10 2017
%F A082173 a(n) = hypergeom([1 - n, -n], [2], 11). - _Peter Luschny_, Mar 19 2018
%p A082173 A082173_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
%p A082173 for w from 1 to n do a[w] := a[w-1]+11*add(a[j]*a[w-j-1],j=1..w-1)od;
%p A082173 convert(a, list) end: A082173_list(17); # _Peter Luschny_, May 19 2011
%t A082173 Table[SeriesCoefficient[(1+10*x-Sqrt[100*x^2-24*x+1])/(22*x),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 14 2012 *)
%t A082173 a[n_] := Hypergeometric2F1[1 - n, -n, 2, 11];
%t A082173 Table[a[n], {n, 0, 18}] (* _Peter Luschny_, Mar 19 2018 *)
%o A082173 (PARI) a(n)=if(n<1,1,sum(k=0,n,11^k/n*binomial(n,k)*binomial(n,k+1)))
%o A082173 (Magma) [1] cat [&+[11^k*Binomial(n, k)*Binomial(n, k+1)/n:k in [0..n]]:n in [1..18]]; // _Marius A. Burtea_, Jan 22 2020
%o A082173 (SageMath)
%o A082173 def A082173_list(prec):
%o A082173     P.<x> = PowerSeriesRing(ZZ, prec)
%o A082173     return P( (1+10*x-sqrt(100*x^2-24*x+1))/(22*x) ).list()
%o A082173 A082173_list(30) # _G. C. Greubel_, Jan 21 2024
%Y A082173 Cf. A001003, A001263, A007564, A059231, A088617.
%K A082173 nonn
%O A082173 0,3
%A A082173 _Benoit Cloitre_, May 10 2003