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A068179 Product_{i=1..3} (i+x) / Product_{i=1..3} (i-x) = Sum_{n>=0} (a(n)/b(n))*x^n.

%I A068179 #48 Jan 01 2024 14:00:33
%S A068179 1,11,121,971,6721,43331,269641,1648091,9981841,60176051,361921561,
%T A068179 2174145611,13052763361,78340331171,470113403881,2820895001531,
%U A068179 16926014399281,101558020876691,609353931324601,3656141011383851
%N A068179 Product_{i=1..3} (i+x) / Product_{i=1..3} (i-x) = Sum_{n>=0} (a(n)/b(n))*x^n.
%C A068179 If n mod 10 == 1, 2, or 4 then a(n)==0 (mod 11). - _Bruno Berselli_, Aug 26 2011
%H A068179 Vincenzo Librandi, <a href="/A068179/b068179.txt">Table of n, a(n) for n = 0..1000</a>
%H A068179 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (11,-36,36).
%F A068179 b(n) = A026532(2*n-1) for n >= 1.
%F A068179 Lim_{n -> infinity} a(n)/b(n) = 12.
%F A068179 From _Yalcin Aktar_, Aug 10 2011: (Start)
%F A068179 a(n) = 5*2^(n+1) + 6^(n+1) - 5*3^(n+1).
%F A068179 a(n)/b(n) = 12 - 30/2^n + 20/3^n.
%F A068179 General case: lim_{m-->+oo} a_n(m)/b_n(m) = A002378(n) where
%F A068179 Product_{i=1..d} (x+i)/Product_{i=1..d} (i-x) = Sum_{n>=0} (a_d(n)/b_d(n))*x^n) = ((-1)^d) * (1 + Sum_{j>=1} (Sum_{k=1..d} ((-1)^k/k^j) * binomial(2*k,k) * binomial(d+k,2*k)) * x^j). (End)
%F A068179 G.f.: (1+36*x^2)/((1-2*x)*(1-3*x)*(1-6*x)). - _Bruno Berselli_, Aug 26 2011
%F A068179 E.g.f.: 10*exp(2*x) - 15*exp(3*x) + 6*exp(6*x). - _G. C. Greubel_, Nov 10 2018
%t A068179 Table[5*2^(n+1)+6^(n+1)-5*3^(n+1), {n,0,20}] (* _G. C. Greubel_, Nov 10 2018 *)
%t A068179 LinearRecurrence[{11,-36,36},{1,11,121},20] (* _Harvey P. Dale_, Aug 16 2021 *)
%o A068179 (Magma) [5*2^(n+1)+6^(n+1)-5*3^(n+1): n in [0..20]]; // _Vincenzo Librandi_, Aug 29 2011
%o A068179 (PARI) vector(20, n, n--; 5*2^(n+1)+6^(n+1)-5*3^(n+1)) \\ _G. C. Greubel_, Nov 10 2018
%o A068179 (Python) for n in range(0,20): print(5*2**(n+1)+6**(n+1)-5*3**(n+1), end=', ') # _Stefano Spezia_, Nov 12 2018
%Y A068179 Cf. A026532.
%K A068179 nonn,frac,easy
%O A068179 0,2
%A A068179 _Benoit Cloitre_, Mar 12 2002