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A162326
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Let y = y(x) satisfy g(x,y(x)) = 0. The sequence a(n) is the number of terms in the expansion of the divided difference [x0,...,xn]y in terms of bivariate divided differences of g.
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3
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1, 3, 13, 71, 441, 2955, 20805, 151695, 1135345, 8671763, 67320573, 529626839, 4213228969, 33833367963, 273892683573, 2232832964895, 18314495896545, 151037687326755, 1251606057754605, 10416531069771111, 87029307323766681
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| (1 + 3x + 13x^2 + 71x^3 + ...) = (1 + 4x + 20x^2 + 116x^3 + ...) * 1/(1 + x + 4x^2 + 20x^3 + 116x^4 + ...); where A082298 = (1, 4, 20, 116, 740,...). - Gary W. Adamson, Nov 17 2011
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FORMULA
| Let E = N x N \ {(0,0), (0,1)} be a set of pairs of natural numbers. The number of terms a(n) is the coefficient of x^n*y^{n-1} of the generating function
1 - log(1 - \sum_{(s,t) in E} x^s*y^{s+t-1}) = 1 + \sum_{q >= 1} (\sum_{(s,t) in E} x^s*y^{s+t-1})^q / q
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EXAMPLE
| Write [0...n]y for [x0,...,xn]y and [0...s,0...t]g for [x0,...,xs;y0,...,yt]g.
For n = 1 one finds 1 term, [01]y = -[01;1]g/[0;01]g.
For n = 2 one finds 3 terms, [012]y = -[012;2]g/[0;02]g + ([01;12]g[12;2]g)/([0;02]g[1;12]g) - ([0;012]g[01;1]g[12;2]g)/([0;02]g[0;01]g[1;12]g).
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PROG
| (Other) # To be executed in Sage 4.0.2 with Singular 3.0.4 as a backend. def P(n, q): ....E = CartesianProduct(range(n+1), range(n+1)) ....E = [(i, j) for (i, j) in E \ ..............if ((i, j) != (0, 0) and (i, j) != (0, 1) \ ..............and i + j <= n and 2*i + j - 1 <= 2*n - q)] . ....return sum([X^s * Y^(s+t-1) for (s, t) in E]) . R.<X, Y> = PolynomialRing(ZZ, 2) . n = 11 h = expand(1 + sum([((P(n, q))^q)/q for q in range(1, 2*n)])) for k in range(1, n+1): ....print k, h.coefficient({X:k, Y:k-1})
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CROSSREFS
| Cf. A172003, which is a generalization to bivariate implicit functions. Cf. A003262, which is the analogous sequence for implicit derivatives, and A172004 for its generalization to bivariate implicit functions.
Cf. A082298
Sequence in context: A024337 A001495 A198447 * A122455 A126390 A167894
Adjacent sequences: A162323 A162324 A162325 * A162327 A162328 A162329
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KEYWORD
| nonn
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AUTHOR
| Georg Muntingh (georg.muntingh(AT)gmail.com), Jul 01 2009
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EXTENSIONS
| Edited by Georg Muntingh (georg.muntingh(AT)gmail.com), Jan 22 2010
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