cp's OEIS Frontend

A172004 Let y = y(u,v) be implicitly defined by g(u,v,y(u,v)) = 0. Read as a triangle by rows k = 1,2,..., the sequence represents the number of terms a(i,k-i) in the expansion of the partial derivatives d^k y/du^i dv^{k-i} in terms of partial derivatives of g.

%I A172004 #14 Mar 06 2020 09:27:05
%S A172004 1,1,3,4,3,9,15,15,9,24,47,59,47,24,61,136,195,195,136,61,145,360,580,
%T A172004 663,580,360,145,333,904,1586,2032,2032,1586,904,333,732,2152,4077,
%U A172004 5684,6350,5684,4077,2152,732,1565,4927,9948,14938,18123,18123,14938,9948
%N A172004 Let y = y(u,v) be implicitly defined by g(u,v,y(u,v)) = 0. Read as a triangle by rows k = 1,2,..., the sequence represents the number of terms a(i,k-i) in the expansion of the partial derivatives d^k y/du^i dv^{k-i} in terms of partial derivatives of g.
%C A172004 The sequence starts with a(1,0),a(0,1),a(2,0),a(1,1),a(0,2),a(3,0),...
%C A172004 The subsequences a(1,0),a(2,0),a(3,0),... and a(0,1),a(0,2),a(0,3),... coincide with the sequence A003262, which is the corresponding sequence for univariate implicit functions.
%H A172004 Wilde, T., <a href="http://arxiv.org/abs/0805.2674">Implicit higher derivatives and a formula of Comtet and Fiolet</a>, preprint arXiv:0805.2674 [math.CO], 2008.
%F A172004 Let E = N^3 \ {(0,0,0), (0,0,1)} be a set of triples of natural numbers. The number of terms a(m,n) is the coefficient of u^m * v^n * y^{m+n-1} in Product_{(r,s,t) in E} (1 - u^r * v^s * y^{r+s+t-1})^{-1}.
%e A172004 The formulas dy/du = -g_u/g_y,
%e A172004 d^2y/du^2 = -g_yy g_u^2/g_y^3 + 2g_uy g_u/g_y^2 - g_uu/g_y,
%e A172004 d^2y/dudv = -2g_yy g_u g_v / g_y^3 + g_uy g_v/g_y^2 + g_vy g_u/g_y^3 - g_uv/g_y
%e A172004 imply that a(1,0) = 1, a(2,0) = 3, and a(1,1) = 4.
%o A172004 (Sage) # Upon executing the following code in Sage 4.2 (using Singular as a backend), it
%o A172004 # computes the number of terms a(n1,n2) and stores it in the entry A[n1][n2] of the
%o A172004 # double list A.
%o A172004 N = 9
%o A172004 E1 = N
%o A172004 E2 = N
%o A172004 p = [[[0 for i1 in range(E1+1)] for i2 in range(E2+1)] for j in range(E1 + E2)]
%o A172004 q = [[[0 for i1 in range(E1+1)] for i2 in range(E2+1)] for j in range(E1 + E2)]
%o A172004 for m in range(1, E1 + E2):
%o A172004     for d in range(1, m+1):
%o A172004         quotient, remainder = divmod(m, d)
%o A172004         if remainder == 0:
%o A172004             for i1 in range(quotient + 1 + 1):
%o A172004                 for i2 in range(quotient + 1 - i1 + 1):
%o A172004                     if d*i1 <= E1 and d*i2 <= E2:
%o A172004                         q[m][i1*d][i2*d] += 1/d
%o A172004 for i1 in range(E1 + 1):
%o A172004     for i2 in range(E2 + 1):
%o A172004         p[0][i1][i2] = 1
%o A172004 for n in range(1, E1 + E2):
%o A172004     for s in range(n+1):
%o A172004         for k1 in range(E1+1):
%o A172004             for k2 in range(E2+1):
%o A172004                 for i1 in range(k1 + 1):
%o A172004                     for i2 in range(k2 + 1):
%o A172004                         p[n][k1][k2] += 1/n * s * q[s][k1-i1][k2-i2] * p[n-s][i1][i2]
%o A172004 A = [[ p[n1+n2-1][n1][n2] for n1 in range(E1+1)] for n2 in range(E2+1)]
%Y A172004 Cf. A003262, which is the univariate variant of this sequence.
%Y A172004 Cf. A172003, which is the analogous sequence for implicit divided differences, and A162326 for its univariate variant.
%K A172004 nonn,tabl
%O A172004 1,3
%A A172004 _Georg Muntingh_, Jan 22 2010