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

%I A092145 #24 Feb 05 2024 10:56:02
%S A092145 1,13,283,8667,342969,16671885,962672355,64467073755,4917699360945,
%T A092145 421377918441165,40104072098340075,4200511400073848475,
%U A092145 480454695780380469225,59617988532820945752525,7980059238850231812954675,1146519564522299271411982875
%N A092145 Numerator of I(n) = 2*(Integral_{x=0..1/2} (1+x^2)^n dx).
%C A092145 The denominator is A034910(n+1) = 2^(n-1)*(2*n+2)!/(n+1)!.
%C A092145 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
%H A092145 Robert Israel, <a href="/A092145/b092145.txt">Table of n, a(n) for n = 0..324</a>
%F A092145 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
%F A092145 a(n) = Sum_{k=0..n} binomial(n,k)/(4^k*(2*k+1)). - _G. C. Greubel_, Feb 05 2024
%F A092145 a(n) ~ 2^(n + 1/2) * 5^(n+1) * n^n / exp(n). - _Vaclav Kotesovec_, Feb 05 2024
%e A092145 I(3) = 8667/6720.
%p A092145 f:= n -> simplify(hypergeom([1/2, -n], [3/2], -1/4)*(2*n+2)!*2^(n-1)/(n+1)!):
%p A092145 map(f, [$0..20]); # _Robert Israel_, Nov 07 2016
%t A092145 a[n_]:= (2^(1+3*n)*Gamma[3/2+n]*Hypergeometric2F1[-n,1/2,3/2,-1/4] )/Sqrt[Pi];
%t A092145 Table[a[n], {n, 0, 20}] (* _Gerry Martens_, Aug 09 2015 *)
%o A092145 (Magma) [Numerator(&+[Binomial(n,k)/(4^k*(2*k+1)): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Feb 05 2024
%o A092145 (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
%Y A092145 Cf. A034910.
%K A092145 nonn,frac
%O A092145 0,2
%A A092145 Al Hakanson (hawkuu(AT)excite.com), Mar 31 2004
%E A092145 More terms from _Alois P. Heinz_, Aug 09 2015