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A036909 a(n) = (2/3) * 4^n * binomial(3*n, n). 2
8, 160, 3584, 84480, 2050048, 50692096, 1270087680, 32133218304, 819082035200, 21002987765760, 541167892561920, 13999778090188800, 363391162981023744, 9459706464902840320, 246865719056498950144, 6456334894356662059008 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

REFERENCES

Identity (3.116) in H. W. Gould, Combinatorial Identities, Morgantown, 1972, page 35.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..695

FORMULA

Sum_{k=0..n} binomial(4*n, 2*(n-k))*binomial(k+n, n) = (2/3)*4^n*binomial(3*n, n) = (2/3)*4^n*A005809(n) = 2*4^n*A025174(n).

G.f.: (2/3) * 2F1([1/3, 2/3], [1/2], 27*x) = 2*(cos((1/6)*arccos(1-54*x))/sqrt(1-27*x) - 1) /(3*x). - Harvey P. Dale, Mar 26 2012

D-finite with recurrence n*(2*n-1)*a(n) = 6*(3*n-1)*(3*n-2)*a(n-1). - R. J. Mathar, Feb 08 2021

G.f.: (2/3)*( cos((1/3)*Arcsin(3*sqrt(3*x)))/sqrt(1-27*x) - 1). - G. C. Greubel, Jun 22 2022

MATHEMATICA

Table[2/3 4^n Binomial[3n, n], {n, 20}](* Harvey P. Dale, Mar 26 2012 *)

PROG

(Magma) [(2/3)*4^n*Binomial(3*n, n): n in [1..30]]; // G. C. Greubel, Jun 22 2022

(SageMath) [(2/3)*4^n*binomial(3*n, n) for n in (0..30)] # G. C. Greubel, Jun 22 2022

CROSSREFS

Cf. A005809, A025174.

Sequence in context: A201030 A127369 A228700 * A221077 A052140 A219265

Adjacent sequences:  A036906 A036907 A036908 * A036910 A036911 A036912

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 17 12:30 EDT 2022. Contains 356189 sequences. (Running on oeis4.)