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A098665 a(n) = Sum_{k = 0..n} binomial(n,k) * binomial(n+1,k+1) * 4^k.

%I A098665 #23 May 21 2024 08:46:37
%S A098665 1,6,43,332,2661,21810,181455,1526040,12939145,110413406,947052723,
%T A098665 8157680228,70518067309,611426078346,5315138311383,46308989294640,
%U A098665 404274406256145,3535479068797110,30966952059306555,271616893912241532,2385412594943633781,20973327081776664546
%N A098665 a(n) = Sum_{k = 0..n} binomial(n,k) * binomial(n+1,k+1) * 4^k.
%C A098665 Fifth binomial transform of A098664.
%H A098665 Vincenzo Librandi, <a href="/A098665/b098665.txt">Table of n, a(n) for n = 0..200</a>
%F A098665 G.f.: ((1+3*x)-sqrt(1-10*x+9*x^2))/(8*x*sqrt(1-10*x+9*x^2)).
%F A098665 E.g.f.: exp(5x)*(BesselI(0, 4x)+BesselI(1, 4x)/2).
%F A098665 Recurrence: (n+1)*(2*n-1)*a(n) = 4*(5*n^2-2)*a(n-1) - 9*(n-1)*(2*n+1)*a(n-2). - _Vaclav Kotesovec_, Oct 15 2012
%F A098665 a(n) ~ 9^(n+1)/(4*sqrt(2*Pi*n)). - _Vaclav Kotesovec_, Oct 15 2012
%F A098665 From _Peter Bala_, Jan 07 2022: (Start)
%F A098665 The following formulas assume an offset of 1:
%F A098665 a(n) = (1/4) * Sum_{k = 0..n} binomial(n,k)*A119259(k).
%F A098665 a(n) = (1/4) * Sum_{k = 0..n} binomial(n,k)*binomial(2*n-k-1,n-k)*3^k.
%F A098665 a(n) = (1/4) * [x^n] ((1 + 3*x)/(1 - x))^n.
%F A098665 The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p >= 3 and positive integers n and k. (End)
%F A098665 a(n) = 4^n*hypergeom([-n, -n - 1], [1], 1/4). - _Detlef Meya_, May 21 2024
%t A098665 Table[SeriesCoefficient[((1+3*x)-Sqrt[1-10*x+9*x^2])/(8*x*Sqrt[1-10*x+9*x^2]),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 15 2012 *)
%t A098665 a[n_] := 4^n*HypergeometricPFQ[{-n, -n - 1}, {1}, 1/4]; Flatten[Table[a[n], {n,0,21}]] (* _Detlef Meya_, May 21 2024 *)
%o A098665 (PARI) my(x='x+O('x^66)); Vec(((1+3*x)-sqrt(1-10*x+9*x^2))/(8*x*sqrt(1-10*x+9*x^2))) \\ _Joerg Arndt_, May 12 2013
%Y A098665 Cf. A098664, A119259.
%K A098665 easy,nonn
%O A098665 0,2
%A A098665 _Paul Barry_, Sep 20 2004