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A264717 Central terms of triangle A100326.

%I A264717 #9 Jan 31 2023 03:33:27
%S A264717 1,4,46,626,9094,136792,2102728,32804760,517325270,8225083124,
%T A264717 131614959262,2116988791018,34196629924584,554369366584256,
%U A264717 9014333613083632,146961155561594176,2401364353568376054,39316907672544234028,644861670750937767370
%N A264717 Central terms of triangle A100326.
%H A264717 Reinhard Zumkeller, <a href="/A264717/b264717.txt">Table of n, a(n) for n = 0..500</a>
%F A264717 a(n) = A100326(2*n,n).
%F A264717 a(n) = (6*(1797120*n^8 -13703040*n^7 +42834240*n^6 -70197188*n^5 +63370677*n^4 -29185735*n^3 +4100685*n^2 +1396683*n - 409602)*a(n-1) +3*(3*n-5)*(3*n-7)*(2*n-3)*(n-2)*(1248*n^4 -780*n^3 -1441*n^2 +1419*n -326)*a(n-2))/(16*(n-1)*(2*n-1)*(4*n-3)*(4*n-1)*(1248*n^4 -5772*n^3 +8387*n^2 -3031*n -1158)). - _G. C. Greubel_, Jan 30 2023
%F A264717 a(n) ~ 3^(3*n/2 - 1) * (1 + sqrt(3))^(6*n + 1/2) / (sqrt(Pi*n) * 2^(7*n + 1/2)). - _Vaclav Kotesovec_, Jan 31 2023
%t A264717 a[n_]:= a[n]= If[n<2, 4^n, (6*(1797120*n^8 -13703040*n^7 +42834240*n^6 -70197188*n^5 +63370677*n^4 -29185735*n^3 +4100685*n^2 +1396683*n - 409602)*a[n-1] +3*(3*n-5)*(3*n-7)*(2*n-3)*(n-2)*(1248*n^4 -780*n^3 -1441*n^2 +1419*n -326)*a[n-2])/(16*(n-1)*(2*n-1)*(4*n-3)*(4*n-1)*(1248*n^4 -5772*n^3 +8387*n^2 -3031*n -1158))];
%t A264717 Table[a[n], {n,0,40}] (* _G. C. Greubel_, Jan 30 2023 *)
%o A264717 (Haskell)
%o A264717 a264717 n = a100326 (2 * n) n
%o A264717 (Magma) [n le 2 select 4^(n-1) else ( 6*(1797120*n^8 -28080000*n^7 +189074880*n^6 -715605188*n^5 +1662275017*n^4 -2421570243*n^3 +2154450632*n^2 -1066134220*n +223382400)*Self(n-1) +3*(3*n-8)*(3*n-10)*(2*n-5)*(n-3)*(1248*n^4 -5772*n^3 +8387*n^2 -3031*n -1158)*Self(n-2))/(16*(n-2)*(2*n-3)*(4*n-7)*(4*n-5)*(1248*n^4 -10764*n^3 +33191*n^2 -42113*n +17280)): n in [1..41]]; // _G. C. Greubel_, Jan 30 2023
%o A264717 (SageMath)
%o A264717 def p(n): return 1797120*n^8 -13703040*n^7 +42834240*n^6 -70197188*n^5 +63370677*n^4 -29185735*n^3 +4100685*n^2 +1396683*n - 409602
%o A264717 def q(n): return (3*n-5)*(3*n-7)*(2*n-3)*(n-2)*(1248*n^4 -780*n^3 -1441*n^2 +1419*n -326)
%o A264717 @CachedFunction
%o A264717 def a(n): # a = A264717
%o A264717     if(n<2): return 4^n
%o A264717     else: return (6*p(n)*a(n-1) +3*q(n)*a(n-2))/(16*(n-1)*(2*n-1)*(4*n-3)*(4*n-1)*(1248*n^4 -5772*n^3 +8387*n^2 -3031*n -1158))
%o A264717 [a(n) for n in range(41)] # _G. C. Greubel_, Jan 30 2023
%Y A264717 Cf. A100326.
%K A264717 nonn
%O A264717 0,2
%A A264717 _Reinhard Zumkeller_, Nov 21 2015