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

1, 12, 198, 3672, 71766, 1444392, 29623644, 615614256, 12918175974, 273112332552, 5808412280628, 124127223181776, 2663248527920124, 57334738304731536, 1237861064261885688, 26791929483836768352

Offset: 0

Author

Paul Barry, Feb 01 2006

Keywords

• easy
• nonn

Comments

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.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• M. Abrate, S. Barbero, U. Cerruti, N. Murru, Fixed Sequences for a Generalization of the Binomial Interpolated Operator and for some Other Operators, J. Int. Seq. 14 (2011) # 11.8.1.
• Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

Programs

Mathematica

CoefficientList[Series[1/Sqrt[1-24*x+36*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

PARI

x='x+O('x^50); Vec(1/sqrt(1-24*x+36*x^2)) \\ G. C. Greubel, Mar 18 2017

PARI

a(n)=pollegendre(n,2)*6^n \\ Charles R Greathouse IV, Mar 18 2017

Formula

G.f.: 1/sqrt(1-24*x+36*x^2).

E.g.f.: exp(12*x)*Bessel_I(0,3*sqrt(12)x).

a(n) = Jacobi_P(n,0,0,sqrt(4))*(3*sqrt(4))^n.

a(n) = 3^n*A069835(n).

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

a(n) ~ sqrt(18+12*sqrt(3))*(12+6*sqrt(3))^n/(6*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 19 2012