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%I #39 May 21 2024 04:06:45
%S 1,8,94,1232,16966,240368,3468844,50712992,748553926,11131168688,
%T 166498969924,2502416381792,37759888297756,571681667171168,
%U 8679980422677784,132116085646644032,2015249400937940806
%N a(n) = 2^n*P_n(4), 2^n times the Legendre polynomial of order n at 4.
%C Central coefficients of (1+8x+15x^2)^n. 2^n*LegendreP(n,k) yields the central coefficients of (1+2kx+(k^2-1)x^2)^n, with g.f. 1/sqrt(1-4kx+4x^2).
%C 16th binomial transform of 2^n*LegendreP(n,-4) = (-1)^n*A098269(n). - _Paul Barry_, Sep 03 2004
%C Diagonal of rational functions 1/(1 + x + 3*y + x*z - 2*x*y*z), 1/(1 - x + y + 3*x*z - 2*x*y*z), 1/(1 - x - x*y - 3*y*z - 2*x*y*z). - _Gheorghe Coserea_, Jul 03 2018
%H Vincenzo Librandi, <a href="/A098269/b098269.txt">Table of n, a(n) for n = 0..200</a>
%H Hacène Belbachir and Abdelghani Mehdaoui, <a href="https://doi.org/10.2989/16073606.2020.1729269">Recurrence relation associated with the sums of square binomial coefficients</a>, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
%H Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Szalay/szalay42.html">Diagonal Sums in the Pascal Pyramid, II: Applications</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
%F G.f.: 1/sqrt(1-16x+4x^2).
%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n, k)*binomial(2(n-k), n)*4^(n-2k).
%F E.g.f.: exp(8*x)*BesselI(0, 2*sqrt(15)*x), cf. A084770. - _Vladeta Jovovic_, Sep 01 2004
%F a(n) = Sum_{k=0..n} binomial(n,k)^2 * 3^k * 5^(n-k). - _Paul D. Hanna_, Sep 29 2012
%F D-finite with recurrence: n*a(n) = 8*(2*n-1)*a(n-1) - 4*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 14 2012
%F a(n) ~ sqrt(450+120*sqrt(15))*(8+2*sqrt(15))^n/(30*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 14 2012
%F a(n) = 3^n*hypergeom([-n, -n], [1], 5/3) = 5^n*hypergeom([-n, -n], [1], 3/5). - _Detlef Meya_, May 21 2024
%t Table[SeriesCoefficient[1/Sqrt[1-16*x+4*x^2],{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 14 2012 *)
%t a[n_] := 3^n*HypergeometricPFQ[{-n, -n}, {1}, 5/3]; Flatten[Table[a[n], {n,0,16}]] (* _Detlef Meya_, May 21 2024 *)
%o (PARI) a(n)=pollegendre(n,4)<<n \\ _Charles R Greathouse IV_, Oct 24 2011
%o (PARI) {a(n)=sum(k=0, n, binomial(n, k)^2*3^k*5^(n-k))} \\ _Paul D. Hanna_, Sep 29 2012
%Y Cf. A069835, A084773.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Sep 01 2004