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A006588 a(n) = 4^n*(3*n)!/((2*n)!*n!). 4
1, 12, 240, 5376, 126720, 3075072, 76038144, 1905131520, 48199827456, 1228623052800, 31504481648640, 811751838842880, 20999667135283200, 545086744471535616, 14189559697354260480, 370298578584748425216, 9684502341534993088512, 253765034617761850982400 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
REFERENCES
W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 35.
The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972; Eq. 3.115, page 35.
LINKS
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
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
MATHEMATICA
Table[4^n*(3*n)!/((2*n)!*n!), {n, 0, 20}] (* Erich Friedman, Mar 22 2008 *)
PROG
(PARI) a(n) = 4^n*(3*n)!/((2*n)!*n!) \\ P L Patodia (pannalal(AT)usa.net), Feb 24 2007
(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
(Magma) [4^n*Factorial(3*n)/(Factorial(2*n)* Factorial(n)): n in [0..20]]; // Vincenzo Librandi, Oct 01 2018
CROSSREFS
Sequence in context: A012303 A119837 A012538 * A009150 A009080 A002166
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved

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Last modified April 24 22:17 EDT 2024. Contains 371964 sequences. (Running on oeis4.)