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A306290
a(n) = 1/(Integral_{x=0..1} (x^3 - x^4)^n dx).
0
1, 20, 252, 2860, 30940, 325584, 3364900, 34337160, 347103900, 3483301360, 34754081648, 345120260940, 3413758188932, 33655718658800, 330869721936600, 3244839440755920, 31754250910172700, 310165459118369712, 3024542552887591120, 29449493278116018800, 286360607519186119920
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