OFFSET
0,2
FORMULA
a(n) = 1/Beta(3*n+1,n+1) = (4*n+1)!/(n! * (3*n)!).
a(n) = Sum_{k = 0..n} (-1)^(n+k) * (3*n + 2*k + 1)*binomial(3*n+k, k). This is the particular case m = 1 of the identity Sum_{k = 0..m*n} (-1)^k * (3*n + 2*k + 1) * binomial(3*n+k, k) = (-1)^(m*n) * (m*n + 1) * binomial((m+3)*n+1, 3*n). - Peter Bala, Nov 02 2024
MATHEMATICA
Table[1/Beta[3*n+1, n+1], {n, 0, 20}]
PROG
(PARI) vector(20, n, n--; (4*n+1)!/(n!*(3*n)!))
(Magma) [Factorial(4*n+1)/(Factorial(n)*Factorial(3*n)): n in [0..20]];
(Sage) [1/beta(3*n+1, n+1) for n in range(20)]
(GAP) List([0..30], n -> Factorial(4*n+1)/(Factorial(n)*Factorial(3*n)));
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
G. C. Greubel, Feb 03 2019
STATUS
approved