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 A181933 #31 Nov 11 2024 20:30:04
%S A181933 0,1,-3,9,-30,106,-385,1421,-5304,19966,-75658,288222,-1102790,
%T A181933 4234868,-16312773,63003869,-243896960,946066678,-3676303578,
%U A181933 14308370014,-55768166380,217640082188,-850345208538,3325907590274,-13020993588680
%N A181933 a(n) = Sum_{k=0..n} binomial(n+k,k)*sin(Pi*(n+k)/2).
%H A181933 G. C. Greubel, <a href="/A181933/b181933.txt">Table of n, a(n) for n = 0..1000</a>
%F A181933 G.f.: (1/2)*(sqrt(4*x+1)*(1+x)-3*x-1)/(sqrt(4*x+1)*(x^2+3*x+1)-4*x^2-5*x-1). - _Vladimir Kruchinin_, Mar 28 2016
%F A181933 a(n) ~ (-1)^(n+1) *2^(2*n+1) / (5*sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 28 2016
%F A181933 Conjecture: +2*n*a(n) +8*n*a(n-1) +(-n+20)*a(n-2) +5*(-n+4)*a(n-3) +2*(-2*n+5)*a(n-4)=0. - _R. J. Mathar_, Jun 14 2016
%t A181933 f[n_] := Sum[ Binomial[n + k, k] Sin[Pi (n + k)/2], {k, 0, n}]; Array[f, 25, 0]
%o A181933 (Maxima)
%o A181933 makelist(coeff(taylor(1/2*(sqrt(4*x+1)*(1+x)-3*x-1)/(sqrt(4*x+1)*(x^2+3*x+1)-4*x^2-5*x-1),x,0,20),x,n),n,0,20); /* _Vladimir Kruchinin_, Mar 28 2016 */
%o A181933 (PARI) x='x+O('x^50); concat([0], Vec((1/2)*(sqrt(4*x+1)*(1+x)-3*x-1)/(sqrt(4*x+1)*(x^2+3*x+1)-4*x^2-5*x-1))) \\ _G. C. Greubel_, Mar 24 2017
%Y A181933 Cf. A026641, A176332.
%K A181933 sign
%O A181933 0,3
%A A181933 _Robert G. Wilson v_, Apr 02 2012