cp's OEIS Frontend

A161199 Numerators in expansion of (1-x)^(-5/2).

%I A161199 #30 Sep 24 2024 03:10:45
%S A161199 1,5,35,105,1155,3003,15015,36465,692835,1616615,7436429,16900975,
%T A161199 152108775,339319575,1502700975,3305942145,115707975075,251835004575,
%U A161199 1091285019825,2354878200675,20251952525805,43397041126725,185423721177825,395033145117975
%N A161199 Numerators in expansion of (1-x)^(-5/2).
%H A161199 Harvey P. Dale, <a href="/A161199/b161199.txt">Table of n, a(n) for n = 0..1000</a>
%F A161199 a(n) = numerator(((3 + 8*n + 4*n^2)/3)*binomial(2*n,n)/(4^n)).
%F A161199 a(n) = denominator((3/2)*Integral_{x=0..1} x^n*sqrt(1-x) dx), where the integral is sqrt(Pi)*n!/Gamma(n+5/2) = n!/( (n+3/2)*(n+1/2)*(n-1/2)*...*(1/2)). - _Groux Roland_, Feb 23 2011
%t A161199 Numerator[CoefficientList[Series[(1-x)^(-5/2),{x,0,30}],x]] (* or *) Numerator[Table[(4n^2+8n+3)/3 Binomial[2n,n]/4^n,{n,0,30}]] (* _Harvey P. Dale_, Oct 15 2011 *)
%o A161199 (Magma)
%o A161199 A161199:= func< n | Numerator( Binomial(n+3,3)*Catalan(n+2)/2^(2*n+1) ) >;
%o A161199 [A161199(n): n in [0..30]]; // _G. C. Greubel_, Sep 24 2024
%o A161199 (SageMath)
%o A161199 def A161199(n): return numerator((-1)^n*binomial(-5/2,n))
%o A161199 [A161199(n) for n in range(31)] # _G. C. Greubel_, Sep 24 2024
%Y A161199 Cf. A161198 (triangle for (1-x)^((-1-2*n)/2) for all values of n).
%Y A161199 Cf. A046161 (denominators for (1-x)^(-5/2)).
%Y A161199 Numerators of [x^n]( (1-x)^(p/2) ): A161202 (p=5), A161200 (p=3), A002596 (p=1), A001790 (p=-1), A001803 (p=-3), this sequence (p=-5), A161201 (p=-7).
%K A161199 easy,nonn,frac
%O A161199 0,2
%A A161199 _Johannes W. Meijer_, Jun 08 2009