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A115865 a(n) = Legendre_P(n,2)*6^n.

%I A115865 #26 Jan 30 2020 21:29:15
%S A115865 1,12,198,3672,71766,1444392,29623644,615614256,12918175974,
%T A115865 273112332552,5808412280628,124127223181776,2663248527920124,
%U A115865 57334738304731536,1237861064261885688,26791929483836768352
%N A115865 a(n) = Legendre_P(n,2)*6^n.
%C A115865 Central coefficients of (1+12*x+27*x^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 A115865 Vincenzo Librandi, <a href="/A115865/b115865.txt">Table of n, a(n) for n = 0..200</a>
%H A115865 M. Abrate, S. Barbero, U. Cerruti, N. Murru, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barbero/barbero9.html"> Fixed Sequences for a Generalization of the Binomial Interpolated Operator and for some Other Operators</a>, J. Int. Seq. 14 (2011) # 11.8.1.
%H A115865 Hacène Belbachir, Abdelghani Mehdaoui, 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 A115865 G.f.: 1/sqrt(1-24*x+36*x^2).
%F A115865 E.g.f.: exp(12*x)*Bessel_I(0,3*sqrt(12)x).
%F A115865 a(n) = Jacobi_P(n,0,0,sqrt(4))*(3*sqrt(4))^n.
%F A115865 a(n) = 3^n*A069835(n).
%F A115865 D-finite with recurrence: n*a(n) +12*(1-2*n)*a(n-1) +36*(n-1)*a(n-2)=0. - _R. J. Mathar_, Nov 14 2011
%F A115865 a(n) ~ sqrt(18+12*sqrt(3))*(12+6*sqrt(3))^n/(6*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 19 2012
%t A115865 CoefficientList[Series[1/Sqrt[1-24*x+36*x^2], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 19 2012 *)
%o A115865 (PARI) x='x+O('x^50); Vec(1/sqrt(1-24*x+36*x^2)) \\ _G. C. Greubel_, Mar 18 2017
%o A115865 (PARI) a(n)=pollegendre(n,2)*6^n \\ _Charles R Greathouse IV_, Mar 18 2017
%K A115865 easy,nonn
%O A115865 0,2
%A A115865 _Paul Barry_, Feb 01 2006