A003262 - OEIS

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.
(Formerly M2791)

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; text; internal format)

OFFSET

1,2

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.

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

EXAMPLE

(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.

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 *)

PROG

(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

End If

For j = 0 To E

p(0, j) = 1

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)

For n = 1 To E

Debug.Print p(n - 1, n)

End Sub

CROSSREFS

Cf. A098504.

Cf. A172004 (generalization to bivariate implicit functions).

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

Cf. A172003 (bivariate variant).

Sequence in context: A084858 A228820 A335470 * A189162 A079282 A117585

Adjacent sequences: A003259 A003260 A003261 * A003263 A003264 A003265

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved