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 A182901 #7 Jul 22 2022 12:11:00 %S A182901 1,1,2,4,8,17,36,78,171,379,848,1912,4341,9915,22767,52526,121698, %T A182901 283043,660579,1546556,3631261,8548643,20174093,47716388,113095740, %U A182901 268575321,638954183,1522668500,3634346039,8687404327,20794957839,49841956726,119610395745 %N A182901 Number of weighted lattice paths in B(n) having no valleys. The members of B(n) are paths of weight n that start at (0,0), end on but never go below the horizontal axis, and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps. A valley is a (1,-1)-step followed by a (1,1)-step. %C A182901 a(n)=A182900(n,0). %D A182901 M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306. %F A182901 G.f.: g=g(z) satisfies z^4*(1+z)g^2-(1-z-z^2-z^3)g+1=0. %F A182901 D-finite with recurrence (n+4)*a(n) +(-n-1)*a(n-1) +3*(-n-2)*a(n-2) +(-n-1)*a(n-3) +(-n+2)*a(n-4) +3*(-n+3)*a(n-5) +(-n+2)*a(n-6) +(n-5)*a(n-7)=0. - _R. J. Mathar_, Jul 22 2022 %e A182901 a(3)=4. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and U=(1,1), D=(1,-1), the four paths of weight 3 are hhh, hH, Hh, and UD; none of them has a valley. %p A182901 eq := z^4*(1+z)*g^2-(1-z-z^2-z^3)*g+1 = 0: g := RootOf(eq, g): gser := series(g, z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 32); %Y A182901 A182900. %K A182901 nonn %O A182901 0,3 %A A182901 _Emeric Deutsch_, Dec 15 2010