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 A372115 #24 Apr 24 2024 07:06:55
%S A372115 1,1,4,11,48,174,784,3219,14816,65082,304656,1393854,6617184,31086556,
%T A372115 149336672,714494467,3466785216,16808037474,82244904016,402770823114,
%U A372115 1984987570016,9797722907684,48581811550112,241324198117678,1202874359046464,6006605345531268
%N A372115 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1 + 4*x)^(1/2) * A(x) ).
%F A372115 G.f.: A(x) = 2/(1 + sqrt(1-4*x*sqrt(1+4*x))).
%F A372115 a(n) = Sum_{k=0..n} 4^(n-k) * binomial(2*k,k) * binomial(k/2,n-k)/(k+1).
%F A372115 D-finite with recurrence n*(n-1)*(n+1)*a(n) +2*n*(n-1)*(10*n-23)*a(n-1) +12*(n-1)*(11*n^2-64*n+83)*a(n-2) +24*(4*n^3-90*n^2+352*n-369)*a(n-3) +48*(-64*n^3+528*n^2-1433*n+1290)*a(n-4) +64*(-268*n^3+3090*n^2-11882*n+15255)*a(n-5) +192*(-208*n^3+2928*n^2-13705*n+21345)*a(n-6) -1152*(4*n-25)*(4*n-19)*(2*n-11)*a(n-7)=0. - _R. J. Mathar_, Apr 24 2024
%o A372115 (PARI) my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x*sqrt(1+4*x))))
%o A372115 (PARI) a(n) = sum(k=0, n, 4^(n-k)*binomial(2*k, k)*binomial(k/2, n-k)/(k+1));
%Y A372115 Cf. A372139.
%K A372115 nonn
%O A372115 0,3
%A A372115 _Seiichi Manyama_, Apr 20 2024