A092145 Numerator of I(n) = 2*(Integral_{x=0..1/2} (1+x^2)^n dx).

1, 13, 283, 8667, 342969, 16671885, 962672355, 64467073755, 4917699360945, 421377918441165, 40104072098340075, 4200511400073848475, 480454695780380469225, 59617988532820945752525, 7980059238850231812954675, 1146519564522299271411982875

Offset: 0

Author

Al Hakanson (hawkuu(AT)excite.com), Mar 31 2004

Keywords

• nonn
• frac

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

Examples

			I(3) = 8667/6720.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..324

Crossrefs

Cf. A034910.

Programs

Magma

[Numerator(&+[Binomial(n,k)/(4^k*(2*k+1)): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 05 2024

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 *)

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

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

Extensions

More terms from Alois P. Heinz, Aug 09 2015