A161199 - OEIS

%I #30 Sep 24 2024 03:10:45

%S 1,5,35,105,1155,3003,15015,36465,692835,1616615,7436429,16900975,

%T 152108775,339319575,1502700975,3305942145,115707975075,251835004575,

%U 1091285019825,2354878200675,20251952525805,43397041126725,185423721177825,395033145117975

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

%H Harvey P. Dale, <a href="/A161199/b161199.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = numerator(((3 + 8*n + 4*n^2)/3)*binomial(2*n,n)/(4^n)).

%F 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 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 (Magma)

%o A161199:= func< n | Numerator( Binomial(n+3,3)*Catalan(n+2)/2^(2*n+1) ) >;

%o [A161199(n): n in [0..30]]; // _G. C. Greubel_, Sep 24 2024

%o (SageMath)

%o def A161199(n): return numerator((-1)^n*binomial(-5/2,n))

%o [A161199(n) for n in range(31)] # _G. C. Greubel_, Sep 24 2024

%Y Cf. A161198 (triangle for (1-x)^((-1-2*n)/2) for all values of n).

%Y Cf. A046161 (denominators for (1-x)^(-5/2)).

%Y 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 easy,nonn,frac

%O 0,2

%A _Johannes W. Meijer_, Jun 08 2009