A084773 Coefficients of 1/sqrt(1-12*x+4*x^2); also, a(n) is the central coefficient of (1+6*x+8*x^2)^n.

1, 6, 52, 504, 5136, 53856, 575296, 6225792, 68026624, 748832256, 8291791872, 92255680512, 1030537089024, 11550176206848, 129824329777152, 1462841567576064, 16518691986407424, 186887008999047168, 2117944490818011136, 24038305911245635584, 273199990096465494016

Offset: 0

Author

Paul D. Hanna, Jun 10 2003

Keywords

• nonn

Comments

Number of Delannoy paths from (0,0) to (n,n) with steps U(0,1), H(1,0) and D(1,1) where H and D can choose from two colors each (or where one step is monochrome and the other two are bicolored). - Paul Barry, May 30 2005

2^n*P_n(3), where P_n is the n-th Legendre polynomial. 2^n*LegendreP(n,k) yields the central coefficients of (1 + 2*k*x + (k^2-1)*x^2)^n, with g.f. 1/sqrt(1 -4*k*x +4*x^2) and e.g.f. exp(2*k*x)*BesselI(0, 2*sqrt(k^2-1)*x). - Paul Barry, May 30 2005

Diagonal of rational functions 1/(1 - x - 2*y - 2*x*y), 1/(1 - x - 2*y*z - 2*x*y*z), 1/(1 - 2*x - y*z - 2*x*y*z). - Gheorghe Coserea, Jul 07 2018

Examples

			G.f.: 1/sqrt(1-2*b*x+(b^2-4*c)*x^2) yields central coefficients of (1+b*x+c*x^2)^n.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..939
• Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
• Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

Crossrefs

Sequences of the form 2^n*LegendreP(n, 2*m+1): A000079 (m=0), this sequence (m=1), A098270 (m=2).

Cf. A001850, A052141, A059474.

See A152254 for another interpretation.

Programs

Magma

[2^n*Evaluate(LegendrePolynomial(n), 3): n in [0..40]]; // G. C. Greubel, May 21 2023

Maple

a := n -> 2^n*hypergeom([-n, -n], [1], 2):
seq(simplify(a(n)), n=0..20); # Peter Luschny, May 20 2015

Mathematica

CoefficientList[Series[1/Sqrt[(1-12x+4x^2)],{x,0,20}],x] (* Harvey P. Dale, Dec 13 2017 *)
Table[2^n*LegendreP[n,3], {n,0,40}] (* G. C. Greubel, May 21 2023 *)

PARI

for(n=0,30,t=polcoeff((1+6*x+8*x^2)^n,n,x); print1(t","))

SageMath

[2^n*gen_legendre_P(n,0,3) for n in range(41)] # G. C. Greubel, May 21 2023

Formula

a(n) = 2*A052141(n) = 2^n * A001850(n), n>0.

From Paul Barry, May 30 2005: (Start)

E.g.f.: exp(6*x)*Bessel_I(0, 2*sqrt(8)*x).

a(n) = Sum_{k=0..floor(n/2)} C(n, k)*C(2(n-k), n)*(-1)^k*3^(n-2*k). (End)

D-finite with recurrence: n*a(n) + 6*(-2*n+1)*a(n-1) + 4*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 30 2012

G.f.: G(0)/2, where G(k) = 1 + 1/( 1 - x*(6-2*x)*(2*k+1)/(x*(6-2*x)*(2*k+1) + (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 17 2013

a(n) = 2^n*hyper2F1(-n, -n, 1, 2). - Peter Luschny, May 20 2015

a(n) = A059474(n,n). - Alois P. Heinz, Oct 05 2017

a(n) ~ 2^(n - 5/4) * (1 + sqrt(2))^(2*n + 1) / sqrt(Pi*n). - Vaclav Kotesovec, Jul 11 2018