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A121725 Generalized central coefficients for k=3. 5
1, 1, 10, 19, 190, 442, 4420, 11395, 113950, 312814, 3128140, 8960878, 89608780, 264735892, 2647358920, 8006545891, 80065458910, 246643289830, 2466432898300, 7711583225338, 77115832253380, 244082045341036, 2440820453410360, 7805301802531534 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Hankel transform of a(n) is 9^binomial(n+1,2). Case k=3 of T(n,k)=2*k^2*(2k)^n*INT(x^n*sqrt(1-x^2)/(1+k^2-2kx),x,-1,1)/pi. T(n,k) has Hankel transform (k^2)^binomial(n+1,2). k=1 corresponds to C(n,floor(n/2)).

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

LINKS

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

FORMULA

a(n) = 18*6^n*INT(x^n*sqrt(1-x^2)/(10-6x),x,-1,1)/Pi.

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

a(n) = Sum_{k, 0<=k<=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)=0. - R. J. Mathar, Nov 26 2012

a(n) ~ (4+(-1)^n) * 2^(n-7/2) * 3^(n+2) / (n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Feb 13 2014

MATHEMATICA

CoefficientList[Series[(1-Sqrt[1-4*9*x^2])/(2*9*x^2)/(1-x*(1-Sqrt[1-4*9*x^2])/(2*9*x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

CROSSREFS

Sequence in context: A299575 A073222 A110463 * A110368 A006050 A045646

Adjacent sequences:  A121722 A121723 A121724 * A121726 A121727 A121728

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Aug 17 2006

EXTENSIONS

More terms from Vincenzo Librandi, Feb 15 2014

STATUS

approved

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Last modified October 21 12:36 EDT 2018. Contains 316419 sequences. (Running on oeis4.)