OFFSET
0,2
COMMENTS
The denominator is A034910(n+1) = 2^(n-1)*(2*n+2)!/(n+1)!.
The terms in the sequence are numerators of unreduced fractions. They equal the value of the integral multiplied by b(n). The reduced fractions are 1, 13/12, 283/240, 2889/2240, 114323/80640, 1111459/709632, 21392719/12300288 etc. - R. J. Mathar, Nov 24 2008
LINKS
Robert Israel, Table of n, a(n) for n = 0..324
FORMULA
a(n) = (2^(3*n+1)*Gamma(n+3/2)/sqrt(Pi))*Hypergeometric2F1([-n, 1/2], [3/2], -1/4). - Gerry Martens, Aug 09 2015
a(n) = Sum_{k=0..n} binomial(n,k)/(4^k*(2*k+1)). - G. C. Greubel, Feb 05 2024
a(n) ~ 2^(n + 1/2) * 5^(n+1) * n^n / exp(n). - Vaclav Kotesovec, Feb 05 2024
EXAMPLE
I(3) = 8667/6720.
MAPLE
f:= n -> simplify(hypergeom([1/2, -n], [3/2], -1/4)*(2*n+2)!*2^(n-1)/(n+1)!):
map(f, [$0..20]); # Robert Israel, Nov 07 2016
MATHEMATICA
a[n_]:= (2^(1+3*n)*Gamma[3/2+n]*Hypergeometric2F1[-n, 1/2, 3/2, -1/4] )/Sqrt[Pi];
Table[a[n], {n, 0, 20}] (* Gerry Martens, Aug 09 2015 *)
PROG
(Magma) [Numerator(&+[Binomial(n, k)/(4^k*(2*k+1)): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 05 2024
(SageMath) [numerator(sum(binomial(n, k)/(4^k*(2*k+1)) for k in range(n+1))) for n in range(31)] # G. C. Greubel, Feb 05 2024
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Al Hakanson (hawkuu(AT)excite.com), Mar 31 2004
EXTENSIONS
More terms from Alois P. Heinz, Aug 09 2015
STATUS
approved