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%I #46 Feb 27 2024 05:54:08
%S 1,1,3,11,41,153,573,2157,8163,31043,118559,454479,1747771,6740059,
%T 26055459,100939779,391785129,1523230569,5931153429,23126146629,
%U 90282147849,352846964649,1380430179489,5405662979649,21186405207549,83101804279101,326199124351701
%N a(n) = b(n) - Sum_{j=0..n-1} b(n) with b(n) = binomial(2*n, n).
%H Indranil Ghosh, <a href="/A281593/b281593.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = [x^n] (2*x-1)/(sqrt(1-4*x)*(x-1)).
%F a(n) = binomial(2*n,n)*(1+hypergeom([1,n+1/2],[n+1],4))+I/sqrt(3).
%F a(n+1) = a(n) + 2*n*Catalan(n).
%F a(n) ~ (4/3)*4^n/sqrt((8*n+2)*Pi/2).
%F D-finite with recurrence n*a(n) +(-7*n+6)*a(n-1) +2*(7*n-13)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - _R. J. Mathar_, Jul 27 2022
%F E.g.f.: exp(2*x)*BesselI(0,2*x) - exp(x)*integral( BesselI(0,2*x)*exp(x) ) dx. - _Mélika Tebni_, Feb 27 2024
%p b := n -> binomial(2*n, n): s := n -> add(b(j), j=0..n):
%p a := n -> b(n) - s(n-1): seq(a(n), n=0..26);
%p # second program:
%p A281593 := series(exp(2*x)*BesselI(0, 2*x) - exp(x)*int(BesselI(0, 2*x)*exp(x), x), x = 0, 27): seq(n!*coeff(A281593, x, n), n=0..26); # _Mélika Tebni_, Feb 27 2024
%t a[n_] = Binomial[2n,n](1+Hypergeometric2F1[1,n+1/2,n+1,4])+I/Sqrt[3];
%t Table[Simplify[a[n]],{n,0,17}]
%t CoefficientList[Series[(2x -1)/((x -1) Sqrt[(1 -4x)]), {x, 0, 26}], x] (* _Robert G. Wilson v_, Feb 25 2017 *)
%t a[0]=1; a[n_]:=a[n-1] + 2*(n-1)*CatalanNumber[n-1];Table[a[n],{n,0,26}] (* _Indranil Ghosh_, Mar 03 2017 *)
%o (Sage)
%o def A():
%o a = b = c = 1
%o yield 1
%o while True:
%o yield a
%o c = (c * (4 * b - 2)) // (b + 1)
%o a += 2 * b * c
%o b += 1
%o a = A(); print([next(a) for _ in (0..25)]) # _Peter Luschny_, Feb 25 2017
%o (PARI) a(n) = binomial(2*n,n)-sum(j=0,n-1,binomial(2*j,j)); /* or */
%o c(n) = binomial(2*n,n)/(n+1);
%o a(n) = if(n==0,1,a(n-1) + 2*(n-1)*c(n-1)); \\ _Indranil Ghosh_, Mar 03 2017
%o (Python)
%o import math
%o def C(n,r): return f(n)/f(r)/f(n-r)
%o def A281593(n):
%o s=0
%o for j in range(0,n):
%o s+=C(2*j,j)
%o return C(2*n,n)-s # _Indranil Ghosh_, Mar 03 2017
%Y A279561(n) = (a(n)+1)/2.
%Y A057552(n) = (a(n+2)-1)/2.
%Y A162551(n) = a(n+1)-a(n).
%Y Cf. A000984, A006134, A279557.
%K nonn
%O 0,3
%A _Peter Luschny_, Feb 25 2017
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