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 A363933 #14 Aug 01 2023 12:16:24
%S A363933 1,1,2,5,14,40,119,361,1113,3476,10971,34919,111949,361100,1171130
%N A363933 Number of polynomials P(x,y) with nonnegative integer coefficients such that P(x,y) == 1 (mod x+y-1) and P(1,1) = n.
%C A363933 The definition was originally used in A279196, which however happened to additionally require the quotient Q(x,y) = (P(x,y)-1) / (x+y-1) to have nonnegative coefficients as well. The current sequence allows these coefficients be negative. Hence a(n) >= A279196(n).
%C A363933 Let Q_d(x,y) be the homogeneous part of Q(x,y) of degree d, and c_d = Q_d(1,1). Then c_0 = 1, c_1, ... form a sequence of nonnegative integers such that c_d <= 2*c_{d-1} and c_0 + c_1 + ... = n-1 (cf. A002572). It follows that Q(x,y) and P(x,y) have degree at most n-2 and at most n-1, respectively.
%e A363933 For n = 5, this sequence but not A279196 accounts for polynomial x^3 + 3xy + y^3 = 1 + (x + y - 1) * (x^2 + y^2 - xy + x + y + 1), explaining why a(5) = 14 while A279196(5) = 13.
%Y A363933 Cf. A279196.
%K A363933 nonn,more
%O A363933 1,3
%A A363933 _Max Alekseyev_, Jun 28 2023