Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #49 Dec 29 2024 04:45:13
%S 1,12,240,5376,126720,3075072,76038144,1905131520,48199827456,
%T 1228623052800,31504481648640,811751838842880,20999667135283200,
%U 545086744471535616,14189559697354260480,370298578584748425216,9684502341534993088512,253765034617761850982400
%N a(n) = 4^n*(3*n)!/((2*n)!*n!).
%D Jonathan Borwein, David Bailey, and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Natick, MA, 2004. See p. 26.
%D William Allen Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 35.
%D Henry W. Gould, Combinatorial Identities, Morgantown, 1972; The right-hand side of a binomial coefficient identity, Eq. 3.115, page 35.
%H Vincenzo Librandi, <a href="/A006588/b006588.txt">Table of n, a(n) for n = 0..710</a>
%H Necdet Batir, <a href="https://doi.org/10.1007/BF02829799">On the series Sum_{k=1..oo} binomial(3k,k)^{-1} k^{-n} x^k</a>, Proc. Indian Acad. Sci. (Math. Sci.), Vol. 115, No. 4 (2005), pp. 371-381; <a href="https://arxiv.org/abs/math/0512310">arXiv preprint</a>, arXiv:math/0512310 [math.AC], 2005.
%H Jonathan M. Borwein and Roland Girgensohn, <a href="https://doi.org/10.1007/s00010-005-2774-x">Evaluations of binomial series</a>, aequationes mathematicae, Vol. 70, No. 1 (2005), pp. 25-36. See p. 31, eq. (37).
%H Marc Le Brun, <a href="/A006577/a006577.pdf">Email to N. J. A. Sloane, Jul 1991</a>.
%F a(n) = Sum_{k=0..n} C(4*n+1, 2*n-2*k)*C(n+k, k) = 4^n*C(3*n, n).
%F a(n) ~ (1/2)*3^(1/2)*Pi^(-1/2)*n^(-1/2)*3^(3*n)*{1 - (7/72)*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Jun 11 2002
%F a(n) = A013609(3*n, 2*n). - _Johannes W. Meijer_, Aug 22 2013
%F a(n) = [x^n] hypergeom([1/3, 2/3], [1/2], 27*x). - _Peter Luschny_, Sep 30 2018
%F a(n) = Sum_{k = n..3*n} binomial(3*n,k)*binomial(k,n). - _Peter Bala_, Mar 25 2023
%F From _Amiram Eldar_, Dec 07 2024:
%F a(n) = 4^n * binomial(3*n, n) = A000302(n) * A005809(n).
%F Sum_{n>=1} (-1)^n/a(n) = -1/28 - 3*log(2)/32 + (13/(112*sqrt(7))) * arctan(sqrt(7)/5) (Borwein et al., 2004; Borwein and Girgensohn, 2005; Batir, 2005). (End)
%p A006588 := n->add( binomial(4*n+1,2*n-2*k)*binomial(n+k,k),k=0..n): seq(A006588(n), n=0..15);
%p h := proc(x) hypergeom([1/3, 2/3], [1/2], 27*x) end: ser := series(h(x), x, 20): seq(coeff(ser, x, n), n=0..15); # _Peter Luschny_, Sep 30 2018
%t Table[4^n*(3*n)!/((2*n)!*n!), {n, 0, 20}] (* _Erich Friedman_, Mar 22 2008 *)
%o (PARI) a(n) = 4^n*(3*n)!/((2*n)!*n!) \\ P L Patodia (pannalal(AT)usa.net), Feb 24 2007
%o (PARI) a(n) = sum(k=0,n,binomial(4*n+1,2*n-2*k)*binomial(n+k,k)) \\ P L Patodia (pannalal(AT)usa.net), Feb 24 2007
%o (Magma) [4^n*Factorial(3*n)/(Factorial(2*n)* Factorial(n)): n in [0..20]]; // _Vincenzo Librandi_, Oct 01 2018
%Y Cf. A000302, A005809, A013609.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_