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A279196
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Number of polynomials of the form P(x,y) = 1 + (x+y-1) * Q(x,y) such that P(1,1) = n and both polynomials P and Q have nonnegative integer coefficients.
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2
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1, 1, 2, 5, 13, 36, 102, 295, 864, 2557, 7624, 22868, 68920, 208527, 632987, 1926752, 5878738, 17973523, 55050690, 168881464, 518818523, 1595878573, 4914522147, 15150038699, 46747391412, 144370209690, 446214862158, 1380161749537, 4271808447154, 13230257155092, 40999697820032
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OFFSET
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1,3
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COMMENTS
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Original definition did not have the requirement for Q to have nonnegative coefficients. However, this results in different terms given by A363933. We have a(n) <= A363933(n), which is strict for n >= 5. - Max Alekseyev, Jun 28 2023
In colorful terms, one can view a polynomial as a configuration made of piles of tokens located at lattice points (i>=0, j>=0). One introduces the notion of "degradation of a configuration": to degrade a configuration, choose a nonempty pile of tokens in it, say that at (i,j); remove one token from that pile; then add one token at (i+1,j) and one token at (i,j+1). This is a nondeterministic process. a(n) is the number of distinct configurations one can possibly get after (n-1) degradations of the initial configuration consisting of just one token at (0,0). In this metaphor, the P's are the resulting configurations and the Q's are records of where the tokens have been taken. - Luc Rousseau, Jun 30 2023
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LINKS
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R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] See sequence "D".
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EXAMPLE
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For n = 1 the a(1) = 1 solution is:
1 = 0(x + y - 1) + 1.
For n = 2 the a(2) = 1 solution is:
x + y = (x + y - 1) + 1.
For n = 3 the a(3) = 2 solutions are:
xy + x + y^2 = (y + 1)(x + y - 1) + 1;
xy + y + x^2 = (x + 1)(x + y - 1) + 1.
For n = 4 the a(4) = 5 solutions are:
x^2 + 2xy + y^2 = (x + y + 1)(x + y - 1) + 1;
x^2y + x^2 + xy^2 + y = (xy + x + 1)(x + y - 1) + 1;
x^2y + xy^2 + x + y^2 = (xy + y + 1)(x + y - 1) + 1;
xy^2 + xy + x + y^3 = (y^2 + y + 1)(x + y - 1) + 1;
x^3 + x^2y + xy + y = (x^2 + x + 1)(x + y - 1) + 1.
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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