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%I #19 Sep 08 2019 02:06:07
%S 1,5,30,193,1286,8754,60460,421985,2968902,21019510,149572292,
%T 1068795930,7664092060,55121602436,397464604440,2872406652001,
%U 20799171328070,150869330458830,1096046132412628,7973709600124958,58081342410990516,423551998861478140
%N a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+1,k+1) * 3^k.
%H Vincenzo Librandi, <a href="/A098663/b098663.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: ((1+2*x) - sqrt(1-8*x+4*x^2))/(6*x*sqrt(1-8*x+4*x^2)).
%F E.g.f.: exp(4x)*(BesselI(0, 2*sqrt(3)*x) + BesselI(1, 2*sqrt(3)*x)/sqrt(3)).
%F Recurrence: (n+1)*(2*n-1)*a(n) = 2*(8*n^2-3)*a(n-1) - 4*(n-1)*(2*n+1)*a(n-2). - _Vaclav Kotesovec_, Oct 15 2012
%F a(n) ~ sqrt(12+7*sqrt(3))*(4+2*sqrt(3))^n/(3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 15 2012
%t Table[Sum[Binomial[n,k]Binomial[n+1,k+1]3^k,{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Nov 08 2011 *)
%o (PARI) x='x+O('x^66); Vec(((1+2*x)-sqrt(1-8*x+4*x^2))/(6*x*sqrt(1-8*x+4*x^2))) \\ _Joerg Arndt_, May 12 2013
%Y Fourth binomial transform of A098662.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Sep 20 2004
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