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 A378939 #7 Dec 19 2024 12:17:18
%S A378939 1,2,5,15,54,216,947,4375,21018,103550,520041,2649391,13655190,
%T A378939 71053780,372727751,1968880111,10463765490,55909445082,300160457453,
%U A378939 1618364548591,8759315367894,47574840887024,259215969470139,1416461749625543,7760734001872842,42624971709868054
%N A378939 Number of Schroeder paths of semilength n up to reversal.
%C A378939 A Schroeder path of semilength n is a path from (0,0) to (2n,0) using only steps U = (1,1), H = (2,0) and D = (1,-1). This sequence considers a path and its reversal to be the same.
%H A378939 Andrew Howroyd, <a href="/A378939/b378939.txt">Table of n, a(n) for n = 0..1000</a>
%F A378939 a(n) = (A006318(n) + A110110(n))/2.
%F A378939 G.f.: ( -2*x - sqrt(1 - 6*x + x^2) + sqrt(1 - 6*x^2 + x^4)*(1 + x)/(1 - 2*x - x^2) ) / (4*x).
%e A378939 The a(1)..a(3) paths are:
%e A378939 a(1) = 1: H, UD;
%e A378939 a(2) = 5: HH, UHD, UDUD, UUDD, HUD=UDH;
%e A378939 a(3) = 15: HHH, HUDH, UHHD, UDHUD, UDUDUD, UUHDD, UUDUDD, UUUDDD, HHUD=UDHH, HUHD=UHDH, HUDUD=UDUDH, UHDUD=UDUHD, HUUDD=UDUDH, UHUDD=UUDHD, UDUUDD=UUDDUD.
%o A378939 (PARI) seq(n) = { my(A=O(x^(n+2))); Vec(( -2*x - sqrt(1 - 6*x + x^2 + A) + sqrt(1 - 6*x^2 + x^4 + A)*(1 + x)/(1 - 2*x - x^2) ) / (4*x)) }
%Y A378939 Cf. A006318, A110110, A007123 (similar for Dyck paths), A378941 (similar for Motzkin paths).
%K A378939 nonn
%O A378939 0,2
%A A378939 _Andrew Howroyd_, Dec 19 2024