A121725 Generalized central coefficients for k=3.

1, 1, 10, 19, 190, 442, 4420, 11395, 113950, 312814, 3128140, 8960878, 89608780, 264735892, 2647358920, 8006545891, 80065458910, 246643289830, 2466432898300, 7711583225338, 77115832253380, 244082045341036, 2440820453410360, 7805301802531534

Offset: 0

Author

Paul Barry, Aug 17 2006

Keywords

• easy
• nonn

Comments

Hankel transform of a(n) is 9^binomial(n+1,2). Case k=3 of T(n,k) = (1/Pi)*2*k^2*(2*k)^n*Integral_{x=-1..1} x^n*sqrt(1-x^2)/(1+k^2-2*k*x) dx. T(n,k) has Hankel transform (k^2)^binomial(n+1,2). k=1 corresponds to C(n, floor(n/2)).

Expansion of c(9*x^2)/(1-x*c(9*x^2)), where c(x) is the g.f. of A000108. Reversion of x*(1+x)/(1+2*x+10*x^2). - Philippe Deléham, Nov 09 2007

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000108, A009766, A120730.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-36*x^2))/(18*x^2-x*(1-Sqrt(1-36*x^2))) )); // G. C. Greubel, Nov 07 2022

Mathematica

CoefficientList[Series[(1-Sqrt[1-4*9*x^2])/(2*9*x^2-x*(1-Sqrt[1-4*9*x^2])), {x,0,40}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

SageMath

def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
def A121725(n): return sum(9^(n-k)*A120730(n,k) for k in range(n+1))
[A121725(n) for n in range(41)] # G. C. Greubel, Nov 07 2022

Formula

a(n) = (1/Pi)*18*6^n*Integral_{x=-1..1} x^n*sqrt(1-x^2)/(10-6*x) dx.

a(n) = Sum_{k=0..floor(n/2)} A009766(n-k,k)*3^2k. - Philippe Deléham, Aug 18 2006

a(n) = Sum_{k=0..n} A120730(n,k)*9^(n-k). - Philippe Deléham, Nov 09 2007

Conjecture: (n+1)*a(n) = 10*(n+1)*a(n-1) + 36*(n-2)*a(n-2) - 360*(n-2)*a(n-3). - R. J. Mathar, Nov 26 2012

From Vaclav Kotesovec, Feb 13 2014: (Start)

a(n) ~ (4+(-1)^n) * 2^(n-7/2) * 3^(n+2) / (n^(3/2) * sqrt(Pi)).

G.f.: (1 - sqrt(1 - 36*x^2))/(18*x^2 - x*(1 - sqrt(1 - 36*x^2))). (End)

Extensions

More terms from Vincenzo Librandi, Feb 15 2014