A006588 a(n) = 4^n*(3*n)!/((2*n)!*n!).
1, 12, 240, 5376, 126720, 3075072, 76038144, 1905131520, 48199827456, 1228623052800, 31504481648640, 811751838842880, 20999667135283200, 545086744471535616, 14189559697354260480, 370298578584748425216, 9684502341534993088512, 253765034617761850982400
Offset: 0
References
- Jonathan Borwein, David Bailey, and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Natick, MA, 2004. See p. 26.
- William Allen Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 35.
- Henry W. Gould, Combinatorial Identities, Morgantown, 1972; The right-hand side of a binomial coefficient identity, Eq. 3.115, page 35.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..710
- Necdet Batir, On the series Sum_{k=1..oo} binomial(3k,k)^{-1} k^{-n} x^k, Proc. Indian Acad. Sci. (Math. Sci.), Vol. 115, No. 4 (2005), pp. 371-381; arXiv preprint, arXiv:math/0512310 [math.AC], 2005.
- Jonathan M. Borwein and Roland Girgensohn, Evaluations of binomial series, aequationes mathematicae, Vol. 70, No. 1 (2005), pp. 25-36. See p. 31, eq. (37).
- Marc Le Brun, Email to N. J. A. Sloane, Jul 1991.
Programs
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Magma
[4^n*Factorial(3*n)/(Factorial(2*n)* Factorial(n)): n in [0..20]]; // Vincenzo Librandi, Oct 01 2018
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Maple
A006588 := n->add( binomial(4*n+1,2*n-2*k)*binomial(n+k,k),k=0..n): seq(A006588(n), n=0..15); 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
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Mathematica
Table[4^n*(3*n)!/((2*n)!*n!), {n, 0, 20}] (* Erich Friedman, Mar 22 2008 *) -
PARI
a(n) = 4^n*(3*n)!/((2*n)!*n!) \\ P L Patodia (pannalal(AT)usa.net), Feb 24 2007
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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
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SageMath
def A006588(n): return 4**n*binomial(3*n,n) print([A006588(n) for n in range(41)]) # G. C. Greubel, Aug 27 2025
Formula
a(n) = Sum_{k=0..n} C(4*n+1, 2*n-2*k)*C(n+k, k) = 4^n*C(3*n, n).
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
a(n) = A013609(3*n, 2*n). - Johannes W. Meijer, Aug 22 2013
a(n) = [x^n] hypergeom([1/3, 2/3], [1/2], 27*x). - Peter Luschny, Sep 30 2018
a(n) = Sum_{k = n..3*n} binomial(3*n,k)*binomial(k,n). - Peter Bala, Mar 25 2023
From Amiram Eldar, Dec 07 2024: (Start)
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)
From G. C. Greubel, Aug 27 2025: (Start)
G.f.: (1/(2*(1-27*x))*( cos(t) + cos(2*t) ), where t = (1/3)*arccos(1-54*x).
E.g.f.: hypergeometric2F2([1/3, 2/3], [1/2, 1], 27*x). (End)