login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A132310 a(n) = 3^n*Sum_{ k=0..n } binomial(2*k,k)/3^k. 13
1, 5, 21, 83, 319, 1209, 4551, 17085, 64125, 240995, 907741, 3428655, 12990121, 49370963, 188229489, 719805987, 2760498351, 10615101273, 40920439119, 158106581157, 612166272291, 2374756691313, 9228369037659, 35918537840577 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Simpler definition from N. J. A. Sloane, Jan 21 2009. Colin Mallows and I studied this sequence on Feb 21 1981 in connection with integration over a regular (solid) hexagon.

Hankel transform is A137717. - Paul Barry, Apr 26 2009

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..300

FORMULA

a(n) = C(2n,n) * sum_{k=0..2n} trinomial(n,k)/C(2n,k) where trinomial(n,k) = [x^k] (1 + x + x^2)^n, where [x^k] denotes "coefficient of x^k in ...".

G.f.: A(x) = 1/sqrt(1 - 10*x + 33*x^2 - 36*x^3).

a(n) = sum_{k=0..2n} trinomial(n,k) * k!*(2*n-k)! / (n!)^2.

2*a(n) = sum(A182411(n+1,i), i=0..n). - Bruno Berselli, May 02 2012

Recurrence: n*a(n) = (7*n-2)*a(n-1) - 6*(2*n-1)*a(n-2) . - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 4^(n+1)/sqrt(Pi*n) . - Vaclav Kotesovec, Oct 20 2012

EXAMPLE

a(1) = C(2,1)*(1/1 + 1/2 + 1/1) = 2*(5/2) = 5;

a(2) = C(4,2)*(1/1 + 2/4 + 3/6 + 2/4 + 1/1) = 6*(7/2) = 21;

a(3) = C(6,3)*(1/1 + 3/6 + 6/15 + 7/20 + 6/15 + 3/6 + 1/1) = 20*(83/20) = 83.

2*a(6) = sum(A182411(7,i), i=0..6) = 3432+858+572+572+728+1092+1848 = 9102 = 2*4551. - Bruno Berselli, May 02 2012

MATHEMATICA

CoefficientList[Series[1/Sqrt[1-10*x+33*x^2-36*x^3], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

PROG

(PARI) a(n)=binomial(2*n, n)*sum(k=0, 2*n, polcoeff((1+x+x^2)^n, k)/binomial(2*n, k) )

(PARI) a(n)=sum(k=0, 2*n, polcoeff((1+x+x^2)^n, k) * k!*(2*n-k)! / (n!)^2 )

CROSSREFS

Sequence in context: A221862 A216271 A026017 * A083319 A146041 A146585

Adjacent sequences:  A132307 A132308 A132309 * A132311 A132312 A132313

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 18 2007

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 12 15:11 EST 2019. Contains 329960 sequences. (Running on oeis4.)