A279196 -id:A279196 - cp's OEIS frontend

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.

1, 1, 2, 5, 14, 40, 119, 361, 1113, 3476, 10971, 34919, 111949, 361100, 1171130

Offset: 1

Max Alekseyev, Jun 28 2023

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).

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.

			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.

Cf. A279196.

cp's OEIS Frontend

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.

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