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A003262
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Let y=f(x) satisfy F(x,y)=0. The sequence a(n) is the number of terms in the expansion of d^ny/dx^n in terms of the partial derivatives of F.
(Formerly M2791)
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3
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1, 3, 9, 24, 61, 145, 333, 732, 1565, 3247, 6583, 13047, 25379, 48477, 91159, 168883, 308736, 557335, 994638, 1755909, 3068960, 5313318, 9118049, 15516710, 26198568, 43904123, 73056724, 120750102, 198304922, 323685343
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 175.
L. Comtet and M. Fiolet, Sur les derivees successives d'une fonction implicite. C. R. Acad. Sci. Paris Ser. A 278 (1974), 249-251.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Wilde, T., Implicit higher derivatives and a formula of Comtet and Fiolet, preprint, 2008.
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FORMULA
| The generating function given by Comtet and Fiolet is incorrect.
a(n)=coeff of t^nu^{n-1} in prod_{i,j>=0,(i,j)<>(0,1)}(1-t^iu^{i+j-1})^{-1}. - Tom Wilde (tom(AT)beech84.fsnet.co.uk), Jan 19 2008
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EXAMPLE
| d^2y/dx^2 = -F_xx/F_y + 2*F_xF_xy/F_y^2 -F_x^2F_yy/F_y^3, where F_x denotes partial derivative wrt x, etc. This has three terms, thus a(n)=3
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PROG
| (VBA, from Tom Wilde) Sub Calc_AofN_upto_E()
E = 30
ReDim p(0 To E - 1, 0 To E): ReDim q(0 To E - 1, 0 To E)
For m = 1 To E - 1: For d = 1 To m
If m = d * Int(m / d) Then
For i = 0 To m / d + 1
If d * i <= E Then q(m, i * d) = q(m, i * d) + 1 / d
Next: End If: Next: Next
For j = 0 To E
p(0, j) = 1
Next
For n = 1 To E - 1: For s = 0 To n: For j = 0 To E: For i = 0 To j
p(n, j) = p(n, j) + 1 / n * s * q(s, j - i) * p(n - s, i)
Next: Next: Next: Next
For n = 1 To E
Debug.Print p(n - 1, n)
Next
End Sub
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CROSSREFS
| Cf. A098504.
Contribution from Georg Muntingh (georg.muntingh(AT)gmail.com), Jan 22 2010: (Start)
Cf. A172004, which is a generalization to bivariate implicit functions.
Cf. A162326, which is the analogous sequence for implicit divided differences, and A172003 for its bivariate variant. (End)
Sequence in context: A086796 A034330 A084858 * A189162 A079282 A117585
Adjacent sequences: A003259 A003260 A003261 * A003263 A003264 A003265
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Tom Wilde (tom(AT)beech84.fsnet.co.uk), Jan 19 2008
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