A077595 Numerator of integral from 0 to 1 of (1 + x^2)^n, in lowest terms.

1, 4, 28, 96, 1328, 4672, 33472, 121856, 3597056, 13417472, 33655808, 127508480, 5829259264, 22308732928, 171393728512, 660468137984, 40831182635008, 22589996269568, 175323994652672, 681560447647744, 10614717931323392, 289707123275726848, 2261982330593738752

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

From Fabian Pereyra, Aug 16 2024: (Start)

a(n) = numerator(Sum_{k=0..n} binomial(n,k)/(2*k+1)).

E.g.f.: Sum_{x>=0} a(n)/A001803(n)*x^n/n! = Integral_{z=0..1} e^(x*(1+z^2)) dz. (End)

Example

For n=3 the integral is 96/35, so a(3) = 96.

Mathematica

a[n_] := Numerator[Integrate[(1 + x x)^n, {x, 0, 1}]]
a[n_] := Hypergeometric2F1[-n, 1/2, 3/2, -1]
Table[Numerator[a[n]], {n, 0, 20}] (* Gerry Martens, Aug 09 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, numerator( subst( intformal((1 + x^2)^n), x, 1)))}

Crossrefs

Cf. A076729.

Sequence in context: A296015 A187452 A173296 * A201243 A092712 A202964

Adjacent sequences: A077592 A077593 A077594 * A077596 A077597 A077598

Keyword

nonn

Author

Michael Somos, Nov 06 2002

Status

approved