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Legendre_P(n,2)*4^n.
1

%I #25 Sep 10 2021 22:03:21

%S 1,8,88,1088,14176,190208,2600704,36030464,504047104,7104278528,

%T 100726755328,1435037302784,20526579564544,294599134674944,

%U 4240277467168768,61183611081064448,884741809748967424

%N Legendre_P(n,2)*4^n.

%C Central coefficients of (1+8x+12x^2)^n. In general, Jacobi_P(n,0,0,sqrt(m))(k*sqrt(m))^n=Legendre_P(n,sqrt(m))(k*sqrt(m))^n has g.f. 1/sqrt(1-2*k*m*x+k^2*x^2), e.g.f. exp(k*m*x)Bessel_I(0,k*sqrt(m(m-1))*x) and gives the central coefficients of (1+k*m*x+k^2*(m(m-1)/4)*x^2)^n.

%H Vincenzo Librandi, <a href="/A115864/b115864.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+16x^2);

%F E.g.f.: exp(8x)Bessel_I(0,2*sqrt(12)x);

%F a(n)=Jacobi_P(n,0,0,sqrt(4))*(2*sqrt(4))^n; a(n)=2^n*A069835(n).

%F D-finite with recurrence: n*a(n) +8*(1-2*n)*a(n-1) +16*(n-1)*a(n-2) =0. - _R. J. Mathar_, Nov 16 2011

%F a(n) ~ sqrt(18+12*sqrt(3))*(8+4*sqrt(3))^n/(6*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 19 2012

%F a(n) = 2^n*A069835(n). - _R. J. Mathar_, Jan 20 2020

%t CoefficientList[Series[1/Sqrt[1-16*x+16*x^2], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 19 2012 *)

%o (PARI) a(n)=pollegendre(n,2)<<(2*n) \\ _Charles R Greathouse IV_, Mar 18 2017

%K easy,nonn

%O 0,2

%A _Paul Barry_, Feb 01 2006