A077745 - OEIS

A077745

Numerator of integral_{x=1..2} (x^2-1)^n dx.

1, 4, 38, 582, 12354, 335730, 11127150, 435300390, 19633815810, 1003121039970, 57259773499950, 3611583223860150, 249441581246630850, 18723487284033181650, 1517668796159163197550, 132117536404977132759750

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OFFSET

0,2

COMMENTS

Denominator is (2n+1)!/(n! 2^n).

Note that these fractions are not reduced. The reduced fractions are 1, 4/3, 38/15, 194/35, 4118/315, 22382/693, 247270/3003, 1381906/6435, etc. and lead to a different sequence of numerators. [From R. J. Mathar, Nov 24 2008]

LINKS

Table of n, a(n) for n=0..15.

FORMULA

(-1)^n*(2*n+1)!!*(2*hypergeom([1/2, -n], [3/2], 4)-hypergeom([1/2, -n], [3/2], 1)). - Vladeta Jovovic, Dec 05 2002

E.g.f.: (2/sqrt(1-6*x)-1)/(1+2*x). - Vladeta Jovovic, Dec 14 2003

a(n) ~ 3*(6*n)^n/(sqrt(2)*exp(n)). - Vaclav Kotesovec, Oct 05 2013

EXAMPLE

If n=3 the integral is 194/35, so a(3) = 7!/(3! 2^3) * 194/35 = 582.

MATHEMATICA

a[n_] := (2n+1)!/n!/2^n*Integrate[(x^2-1)^n, {x, 1, 2}]

CROSSREFS

Cf. A076729.