The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #23 Sep 02 2024 17:35:26
%S 1,4,28,96,1328,4672,33472,121856,3597056,13417472,33655808,127508480,
%T 5829259264,22308732928,171393728512,660468137984,40831182635008,
%U 22589996269568,175323994652672,681560447647744,10614717931323392,289707123275726848,2261982330593738752
%N Numerator of integral from 0 to 1 of (1 + x^2)^n, in lowest terms.
%F From _Fabian Pereyra_, Aug 16 2024: (Start)
%F a(n) = numerator(Sum_{k=0..n} binomial(n,k)/(2*k+1)).
%F E.g.f.: Sum_{x>=0} a(n)/A001803(n)*x^n/n! = Integral_{z=0..1} e^(x*(1+z^2)) dz. (End)
%e For n=3 the integral is 96/35, so a(3) = 96.
%t a[n_] := Numerator[Integrate[(1 + x x)^n, {x, 0, 1}]]
%t a[n_] := Hypergeometric2F1[-n, 1/2, 3/2, -1]
%t Table[Numerator[a[n]], {n, 0, 20}] (* _Gerry Martens_, Aug 09 2015 *)
%o (PARI) {a(n) = if( n<0, 0, numerator( subst( intformal((1 + x^2)^n), x, 1)))}
%Y Cf. A076729.
%K nonn
%O 0,2
%A _Michael Somos_, Nov 06 2002