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%I M2791 #46 Oct 16 2023 03:33:41
%S 1,3,9,24,61,145,333,732,1565,3247,6583,13047,25379,48477,91159,
%T 168883,308736,557335,994638,1755909,3068960,5313318,9118049,15516710,
%U 26198568,43904123,73056724,120750102,198304922,323685343
%N 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.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 175.
%D L. Comtet and M. Fiolet, Sur les dérivées successives d'une fonction implicite. C. R. Acad. Sci. Paris Ser. A 278 (1974), 249-251.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Robert G. Wilson v, <a href="/A003262/b003262.txt">Table of n, a(n) for n = 1..500</a>
%H L. Comtet, <a href="/A003262/a003262.pdf">Letter to N. J. A. Sloane, Mar 1974</a>.
%H L. Comtet and M. Fiolet, <a href="/A003262/a003262_1.pdf">Number of terms in an nth derivative</a>, C. R. Acad. Sc. Paris, t. 278 (21 janvier 1974), Serie A- 249-251. (Annotated scanned copy)
%H T. Wilde, <a href="https://arxiv.org/abs/0805.2674">Implicit higher derivatives and a formula of Comtet and Fiolet</a>, arXiv:0805.2674 [math.CO], 2008.
%F The generating function given by Comtet and Fiolet is incorrect.
%F 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
%e (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.
%t 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 *)
%o (VBA)
%o ' Tom Wilde, Jan 19 2008
%o Sub Calc_AofN_upto_E()
%o E = 30
%o ReDim p(0 To E - 1, 0 To E)
%o ReDim q(0 To E - 1, 0 To E)
%o For m = 1 To E - 1
%o For d = 1 To m
%o If m = d * Int(m / d) Then
%o For i = 0 To m / d + 1
%o If d * i <= E Then q(m, i * d) = q(m, i * d) + 1 / d
%o Next
%o End If
%o Next
%o Next
%o For j = 0 To E
%o p(0, j) = 1
%o Next
%o For n = 1 To E - 1
%o For s = 0 To n
%o For j = 0 To E
%o For i = 0 To j
%o p(n, j) = p(n, j) + 1 / n * s * q(s, j - i) * p(n - s, i)
%o Next
%o Next
%o Next
%o Next
%o For n = 1 To E
%o Debug.Print p(n - 1, n)
%o Next
%o End Sub
%Y Cf. A098504.
%Y Cf. A172004 (generalization to bivariate implicit functions).
%Y Cf. A162326 (analogous sequence for implicit divided differences).
%Y Cf. A172003 (bivariate variant).
%K nonn,nice,easy
%O 1,2
%A _N. J. A. Sloane_
%E More terms from Tom Wilde (tom(AT)beech84.fsnet.co.uk), Jan 19 2008
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