Search: id:A003262
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%I A003262 M2791
%S A003262 1,3,9,24,61,145,333,732,1565,3247,6583,13047,25379,48477,91159,168883,
%T A003262 308736,557335,994638,1755909,3068960,5313318,9118049,15516710,26198568,
%U A003262 43904123,73056724,120750102,198304922,323685343
%N A003262 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.
%D A003262 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A003262 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 175.
%D A003262 L. Comtet and M. Fiolet, Sur les derivees successives d'une fonction 
               implicite. C. R. Acad. Sci. Paris Ser. A 278 (1974), 249-251.
%D A003262 Wilde, T., Implicit higher derivatives and a formula of Comtet and Fiolet, 
               preprint, 2008.
%F A003262 The generating function given by Comtet and Fiolet is incorrect.
%F A003262 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
%e A003262 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
%o A003262 (VBA, from Tom Wilde) Sub Calc_AofN_upto_E()
%o A003262 E = 30
%o A003262 ReDim p(0 To E - 1, 0 To E): ReDim q(0 To E - 1, 0 To E)
%o A003262 For m = 1 To E - 1: For d = 1 To m
%o A003262 If m = d * Int(m / d) Then
%o A003262 For i = 0 To m / d + 1
%o A003262 If d * i <= E Then q(m, i * d) = q(m, i * d) + 1 / d
%o A003262 Next: End If: Next: Next
%o A003262 For j = 0 To E
%o A003262 p(0, j) = 1
%o A003262 Next
%o A003262 For n = 1 To E - 1: For s = 0 To n: For j = 0 To E: For i = 0 To j
%o A003262 p(n, j) = p(n, j) + 1 / n * s * q(s, j - i) * p(n - s, i)
%o A003262 Next: Next: Next: Next
%o A003262 For n = 1 To E
%o A003262 Debug.Print p(n - 1, n)
%o A003262 Next
%o A003262 End Sub
%Y A003262 Cf. A098504.
%Y A003262 Sequence in context: A086796 A034330 A084858 this_sequence A079282 A117585 
               A006684
%Y A003262 Adjacent sequences: A003259 A003260 A003261 this_sequence A003263 A003264 
               A003265
%K A003262 nonn,nice,easy
%O A003262 1,2
%A A003262 N. J. A. Sloane (njas(AT)research.att.com).
%E A003262 More terms from Tom Wilde (tom(AT)beech84.fsnet.co.uk), Jan 19 2008

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