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 A097894 #32 Mar 07 2023 11:06:40
%S A097894 1,4,14,44,134,400,1184,3488,10253,30108,88386,259492,762085,2239120,
%T A097894 6582280,19360432,56976859,167774428,494301778,1457104948,4297477252,
%U A097894 12680944960,37436553544,110569987344,326713395019,965775778420
%N A097894 Partial sums of A014531.
%C A097894 a(n) = number of peaks at even height in all Motzkin paths of length n+3. Example: a(2)=4 because in the 21 Motzkin paths of length 5 we have altogether 4 peaks at even height (shown between parentheses): HU(UD)D, U(UD)DH, U(UD)HD, UH(UD)D.
%C A097894 This is a kind of Motzkin transform of A121262 because the substitution x -> x*A001006(x) in the independent variable of the g.f. A121262(x) defines a sequence which is 1,0,0,0 followed by this sequence here. - _R. J. Mathar_, Nov 08 2008
%H A097894 Vincenzo Librandi, <a href="/A097894/b097894.txt">Table of n, a(n) for n = 1..200</a>
%H A097894 Jean-Luc Baril and José Luis Ramírez, <a href="https://arxiv.org/abs/2302.12741">Descent distribution on Catalan words avoiding ordered pairs of Relations</a>, arXiv:2302.12741 [math.CO], 2023.
%H A097894 László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Nemeth/nemeth6.html">The trinomial transform triangle</a>, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also <a href="https://arxiv.org/abs/1807.07109">arXiv:1807.07109</a> [math.NT], 2018.
%F A097894 G.f.: (1-2*x-x^2)/(2*x^3*(1-x)*sqrt(1-2*x-3*x^2))-1/(2*x^3). D-finite with recurrence -(n-1)*(n+3)*a(n) +(n+2)*(3n-1)*a(n-1) +(n-1)*(n+1)*a(n-2) -3*n*(n+1)*a(n-3)=0. - _R. J. Mathar_, Nov 17 2011
%F A097894 a(n) ~ 3^(n+5/2) / (4*sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 01 2014
%p A097894 ser:=series((1-2*z-z^2)/2/z^3/(1-z)/sqrt(1-2*z-3*z^2)-1/2/z^3,z=0,32): seq(coeff(ser,z^n),n=1..28);
%t A097894 CoefficientList[Series[((1-2*x-x^2)/(2*x^3*(1-x)*Sqrt[1-2*x-3*x^2])-1/(2*x^3))/x, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 01 2014 *) (* adapted to the offset by _Vincenzo Librandi_, Feb 13 2014 *)
%o A097894 (PARI) x='x+O('x^30); Vec((1-2*x-x^2)/(2*x^3*(1-x)*sqrt(1-2*x-3*x^2))-1/(2*x^3)) \\ _G. C. Greubel_, Dec 20 2017
%Y A097894 Cf. A014531.
%K A097894 nonn
%O A097894 1,2
%A A097894 _Emeric Deutsch_, Sep 03 2004