This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A372611 #6 May 28 2024 21:13:03
%S A372611 1,7,26,90,310,1082,3844,13892,50950,189130,708876,2677452,10175356,
%T A372611 38863780,149045960,573559240,2213551430,8563950250,33203854460,
%U A372611 128978378620,501839077460,1955475615820,7629823818680,29805375256120,116558646378140,456270710243332
%N A372611 Expansion of (1 + 3*x) / ((1 - 2*x)*sqrt(1 - 4*x)).
%C A372611 Conjecture: For p Pythagorean prime (A002144), a(p) - 7 == 0 (mod p).
%C A372611 Conjecture: For p prime of the form 4*k + 3 (A002145), a(p) + 3 == 0 (mod p).
%F A372611 a(n) = 6*A000984(n) - 5* A029759(n) = binomial(2*n,n) + 5*Sum_{k=0..n-1} 2^(n-k-1)*binomial(2*k,k).
%F A372611 a(n) = 2*a(n-1) + A028322(n) = 2*a(n-1) + binomial(2*n, n) + 3*binomial(2*n-2, n-1) for n >= 1.
%F A372611 a(n) = - 2^(n-1)*5*i + binomial(2*n,n)*(1-5/2*hypergeom([1, n + 1/2], [n + 1], 2)).
%F A372611 a(n) = 3*A082590(n-1) + A082590(n) for n >= 1.
%F A372611 a(n) = (7*A188622(n) - 4*A126966(n))/3.
%F A372611 a(n) = 2*A372239(n) - A372420(n).
%p A372611 a := n -> -2^(n-1)*5*I + binomial(2*n, n)*(1-5/2*hypergeom([1, n+1/2], [n+1], 2)): seq(simplify(a(n)), n = 0 .. 25);
%o A372611 (PARI) my(x='x+O('x^30)); Vec((1 + 3*x) / ((1 - 2*x)*sqrt(1 - 4*x))) \\ _Michel Marcus_, May 07 2024
%Y A372611 Cf. A002144, A002145, A000984, A028322, A029759, A082590, A126966, A188622, A372239, A372420.
%K A372611 nonn
%O A372611 0,2
%A A372611 _Mélika Tebni_, May 07 2024