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%I #11 May 01 2024 13:05:39
%S 1,5,18,62,214,750,2676,9708,35718,132926,499228,1888644,7186876,
%T 27478508,105474216,406182552,1568563014,6071812638,23552366796,
%U 91525132692,356242058004,1388588519268,5419533876696,21176597444712,82834229300124,324326668721100
%N Expansion of (1 + x) / ((1 - 2*x)*sqrt(1 - 4*x)).
%C Conjecture: For p Pythagorean prime (A002144), a(p) - 5 == 0 (mod p).
%C Conjecture: For p prime of the form 4*k + 3 (A002145), a(p) + 1 == 0 (mod p).
%F a(n) = 4*A000984(n) - 3* A029759(n) = binomial(2*n,n) + 3*Sum_{k=0..n-1} 2^(n-k-1)*binomial(2*k,k).
%F a(n) = 2*a(n-1) + A028270(n) = 2*a(n-1) + binomial(2*n, n) + binomial(2*n-2, n-1) for n >= 1.
%F a(n) = - 2^(n-1)*3*i + binomial(2*n,n)*(1-3/2*hypergeom([1,n+1/2],[n + 1],2)).
%F a(n) = A082590(n-1) + A082590(n) for n >= 1.
%F a(n) = (5*A188622(n) - 2*A126966(n)) / 3.
%F D-finite with recurrence n*a(n) -5*n*a(n-1) +2*(n+5)*a(n-2) +4*(2*n-5)*a(n-3)=0. - _R. J. Mathar_, May 01 2024
%p a := n -> -2^(n-1)*3*I + binomial(2*n, n)*(1-3/2*hypergeom([1, n+1/2], [n+1], 2)):
%p seq(simplify(a(n)), n = 0 .. 25);
%o (PARI) my(x='x+O('x^40)); Vec((1 + x) / ((1 - 2*x)*sqrt(1 - 4*x))) \\ _Michel Marcus_, Apr 30 2024
%Y Cf. A002144, A002145, A000984, A028270, A029759, A082590, A126966, A188622.
%K nonn
%O 0,2
%A _Mélika Tebni_, Apr 30 2024
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