|
Search: id:A003262
|
|
|
| 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.
(Formerly M2791)
|
|
+0
2
|
|
| 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; listen)
|
|
|
OFFSET
|
1,2
|
|
|
REFERENCES
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.
Wilde, T., Implicit higher derivatives and a formula of Comtet and Fiolet, preprint, 2008.
|
|
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
|
|
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
|
|
PROGRAM
|
(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
|
|
CROSSREFS
|
Cf. A098504.
Adjacent sequences: A003259 A003260 A003261 this_sequence A003263 A003264 A003265
Sequence in context: A086796 A034330 A084858 this_sequence A079282 A117585 A006684
|
|
KEYWORD
|
nonn,nice,easy
|
|
AUTHOR
|
N. J. A. Sloane (njas(AT)research.att.com).
|
|
EXTENSIONS
|
More terms from Tom Wilde (tom(AT)beech84.fsnet.co.uk), Jan 19 2008
|
|
|
Search completed in 0.002 seconds
|
