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A003262
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Let y=f(x) satisfy F(x,y)=0. a(n) is the number of terms in the expansion of (d/dx)^n y in terms of the partial derivatives of F.
(Formerly M2791)
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4
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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
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OFFSET
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1,2
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 175.
L. Comtet and M. Fiolet, Sur les dérivées 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).
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LINKS
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FORMULA
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The generating function given by Comtet and Fiolet is incorrect.
a(n) = coefficient of t^n*u^(n-1) in Product_{i,j>=0,(i,j)<>(0,1)} (1 - t^i*u^(i+j-1))^(-1). - Tom Wilde (tom(AT)beech84.fsnet.co.uk), Jan 19 2008
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EXAMPLE
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(d/dx)^2 y = -F_xx/F_y + 2*F_x*F_xy/F_y^2 - F_x^2*F_yy/F_y^3, where F_x denotes partial derivative with respect to x, etc. This has three terms, thus a(2)=3.
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MATHEMATICA
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p[_, _] = 0; q[_, _] = 0; e = 30; For[m = 1, m <= e - 1, m++, For[d = 1, d <= m, d++, If[m == d*Floor[m/d], For[i = 0, i <= m/d + 1, i++, If[d*i <= e, q[m, i*d] = q[m, i*d] + 1/d]]]]]; For[j = 0, j <= e, j++, p[0, j] = 1]; For[n = 1, n <= e - 1, n++, For[s = 0, s <= n, s++, For[j = 0, j <= e, j++, For[i = 0, i <= j, i++, p[n, j] = p[n, j] + (1/n)*s*q[s, j - i]*p[n - s, i]]]]]; A003262 = Table[p[n - 1, n], {n, 1, e}](* Jean-François Alcover, after Tom Wilde *)
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PROG
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(VBA)
' Tom Wilde, Jan 19 2008
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
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Cf. A172004 (generalization to bivariate implicit functions).
Cf. A162326 (analogous sequence for implicit divided differences).
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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More terms from Tom Wilde (tom(AT)beech84.fsnet.co.uk), Jan 19 2008
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STATUS
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approved
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