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 A097332 #39 Jan 14 2025 23:09:11
%S A097332 1,2,3,5,9,18,39,90,217,540,1375,3563,9361,24872,66707,180341,490913,
%T A097332 1344380,3701159,10237541,28436825,79288844,221836403,622599626,
%U A097332 1752360041,4945087838,13988490339,39658308815,112666081617
%N A097332 Expansion of (1/(1-x))(1+2x/(1-x+sqrt(1-2x-3x^2))).
%C A097332 Binomial transform of A097331. Binomial transform is A014318. Partial sums of 1+2x/(1-x+sqrt(1-2x-3x^2)) or (1+x+sqrt(1-2x-3x^2))/(1-x+sqrt(1-2x-3x^2)), which is A001006 with an extra leading 1.
%C A097332 Apparently the Motzkin transform of 1, 2, bar(1, -1, -1, 1), where bar() denotes a periodically continued series, as in A057077. - _R. J. Mathar_, Dec 11 2008
%C A097332 Starting with offset 1 = iterates of M * [1,1,0,0,0,...] where M = a tridiagonal matrix with [1,1,1,...] in the main and superdiagonals and [0,1,1,1,...] in the subdiagonal. - _Gary W. Adamson_, Jan 08 2009
%C A097332 Hankel transform is A087960(n) = (-1)^binomial(n+1,2). - _Paul Barry_, Aug 10 2009
%H A097332 Vincenzo Librandi, <a href="/A097332/b097332.txt">Table of n, a(n) for n = 0..200</a>
%H A097332 Emeric Deutsch and Bruce E. Sagan, <a href="https://arxiv.org/abs/math/0407326">Congruences for Catalan and Motzkin numbers and related sequences</a>, arXiv:math/0407326 [math.CO], 2004; J. Num. Theory 117 (2006), 191-215. [See S_n on page 7.]
%F A097332 a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(n, k)*Sum_{i=0..k} Catalan(k-i)*2^i.
%F A097332 G.f.: 1/(1-x-x/(1+x/(1-x+x/(1-x/(1-x-x/(1+x/(1-x+x/(1-x/(1-x-x/(1+... (continued fraction). - _Paul Barry_, Aug 10 2009
%F A097332 Conjecture D-finite with recurrence: (n+1)*a(n) - 3*n*a(n-1) + (-n+5)*a(n-2) + 3*(n-2)*a(n-3) = 0. - _R. J. Mathar_, Nov 26 2012
%F A097332 a(n) ~ 3^(n+3/2) / (4 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 13 2014
%F A097332 0 = a(n)*(9*a(n+1) + 6*a(n+2) - 27*a(n+3) + 12*a(n+4)) + a(n+1)*(-12*a(n+1) + 10*a(n+2) + 12*a(n+3) - 7*a(n+4)) + a(n+2)*(-12*a(n+2) + 14*a(n+3) - 6*a(n+4)) + a(n+3)*(a(n+4)). - _Michael Somos_, May 19 2014
%e A097332 G.f. = 1 + 2*x + 3*x^2 + 5*x^3 + 9*x^4 + 18*x^5 + 39*x^6 + 90*x^7 + 217*x^8 + ...
%t A097332 CoefficientList[Series[1/(1-x)*(1+(2x)/(1-x+Sqrt[1-2x-3x^2])),{x,0,40}],x] (* _Harvey P. Dale_, May 03 2012 *)
%t A097332 a[ n_] := SeriesCoefficient[ (1 + x - Sqrt[1 - 2 x - 3 x^2]) / (2 x (1 - x)), {x, 0, n}]; (* _Michael Somos_, May 19 2014 *)
%o A097332 (PARI) {a(n) = if( n<0, 0, polcoeff( (1 + x - sqrt(1 - 2*x - 3*x^2 + x^2 * O(x^n))) / (2 * x * (1 - x)), n))}; /* _Michael Somos_, May 19 2014 */
%K A097332 easy,nonn
%O A097332 0,2
%A A097332 _Paul Barry_, Aug 05 2004