A306290 a(n) = 1/(Integral_{x=0..1} (x^3 - x^4)^n dx).

1, 20, 252, 2860, 30940, 325584, 3364900, 34337160, 347103900, 3483301360, 34754081648, 345120260940, 3413758188932, 33655718658800, 330869721936600, 3244839440755920, 31754250910172700, 310165459118369712, 3024542552887591120, 29449493278116018800, 286360607519186119920

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Necdet Batir, On certain series involving reciprocals of binomial coefficients, Journal of Classical Analysis, Vol. 2, No. 1 (2013), pp. 1-8.

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

From Amiram Eldar, Dec 09 2024: (Start)

a(n) = (4*n + 1) * binomial(4*n, n) = A016813(n) * A005810(n).

Formulas from Batir (2013):

Sum_{n>=0} 1/a(n) = f(c) = 1.05435362585114283076..., where f(x) = (x*(x^2-1)/(2*(2*x^2+1))) * log(abs((x+1)/(x-1))) + ((x-1)*(x^3+1)/(4*x*(2*x^2+1))) * sqrt(x/(x-2)) * (arctan(sqrt(x/(x-2))) + arctan(((3-x)/(x+1))*sqrt(x/(x-2)))) + ((x+1)*(x^3-1)/(4*x*(2*x^2+1))) * sqrt(x/(x+2)) * (arctan(((x+3)/(x-1))*sqrt(x/(x+2))) - arctan(sqrt(x/(x+2)))), and c = sqrt(1 + (16/sqrt(3))*cos(arctan(sqrt(229/27))/3)).

Sum_{n>=0} (-1)^n/a(n) = f(d) = 0.953648123517883351708..., where f(x) is defined above, and d = sqrt(1 + 16*(2/(3*(9+sqrt(849))))^(1/3) - 2*(2/3)^(2/3)*(9+sqrt(849))^(1/3)). (End)

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

Cf. A002457, A005810, A016813, A090816, A090957, A090969.

Sequence in context: A081143 A218101 A241228 * A075469 A028032 A348055

Adjacent sequences: A306287 A306288 A306289 * A306291 A306292 A306293

Keyword

nonn,easy

Author

G. C. Greubel, Feb 03 2019

Status

approved