A306290 - OEIS

%I #13 Nov 03 2024 09:33:31

%S 1,20,252,2860,30940,325584,3364900,34337160,347103900,3483301360,

%T 34754081648,345120260940,3413758188932,33655718658800,

%U 330869721936600,3244839440755920,31754250910172700,310165459118369712,3024542552887591120,29449493278116018800,286360607519186119920

%N a(n) = 1/(Integral_{x=0..1} (x^3 - x^4)^n dx).

%F a(n) = 1/Beta(3*n+1,n+1) = (4*n+1)!/(n! * (3*n)!).

%F 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

%t Table[1/Beta[3*n+1, n+1], {n, 0, 20}]

%o (PARI) vector(20, n, n--; (4*n+1)!/(n!*(3*n)!))

%o (Magma) [Factorial(4*n+1)/(Factorial(n)*Factorial(3*n)): n in [0..20]];

%o (Sage) [1/beta(3*n+1,n+1) for n in range(20)]

%o (GAP) List([0..30], n -> Factorial(4*n+1)/(Factorial(n)*Factorial(3*n)));

%Y Cf. A002457, A090816, A090957, A090969.

%K nonn,easy,changed

%O 0,2

%A _G. C. Greubel_, Feb 03 2019