A003262 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.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Examples

			(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.
		

References

• 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).

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..500
• L. Comtet, Letter to N. J. A. Sloane, Mar 1974.
• L. Comtet and M. Fiolet, Number of terms in an nth derivative, C. R. Acad. Sc. Paris, t. 278 (21 janvier 1974), Serie A- 249-251. (Annotated scanned copy)
• T. Wilde, Implicit higher derivatives and a formula of Comtet and Fiolet, arXiv:0805.2674 [math.CO], 2008.

Crossrefs

Cf. A098504.

Cf. A172004 (generalization to bivariate implicit functions).

Cf. A162326 (analogous sequence for implicit divided differences).

Cf. A172003 (bivariate variant).

Programs

Mathematica

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 *)

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

Formula

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

Extensions

More terms from Tom Wilde (tom(AT)beech84.fsnet.co.uk), Jan 19 2008