A172004 - OEIS

%I #14 Mar 06 2020 09:27:05

%S 1,1,3,4,3,9,15,15,9,24,47,59,47,24,61,136,195,195,136,61,145,360,580,

%T 663,580,360,145,333,904,1586,2032,2032,1586,904,333,732,2152,4077,

%U 5684,6350,5684,4077,2152,732,1565,4927,9948,14938,18123,18123,14938,9948

%N 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 The sequence starts with a(1,0),a(0,1),a(2,0),a(1,1),a(0,2),a(3,0),...

%C 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 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 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 The formulas dy/du = -g_u/g_y,

%e d^2y/du^2 = -g_yy g_u^2/g_y^3 + 2g_uy g_u/g_y^2 - g_uu/g_y,

%e 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 imply that a(1,0) = 1, a(2,0) = 3, and a(1,1) = 4.

%o (Sage) # Upon executing the following code in Sage 4.2 (using Singular as a backend), it

%o # computes the number of terms a(n1,n2) and stores it in the entry A[n1][n2] of the

%o # double list A.

%o N = 9

%o E1 = N

%o E2 = N

%o p = [[[0 for i1 in range(E1+1)] for i2 in range(E2+1)] for j in range(E1 + E2)]

%o q = [[[0 for i1 in range(E1+1)] for i2 in range(E2+1)] for j in range(E1 + E2)]

%o for m in range(1, E1 + E2):

%o     for d in range(1, m+1):

%o         quotient, remainder = divmod(m, d)

%o         if remainder == 0:

%o             for i1 in range(quotient + 1 + 1):

%o                 for i2 in range(quotient + 1 - i1 + 1):

%o                     if d*i1 <= E1 and d*i2 <= E2:

%o                         q[m][i1*d][i2*d] += 1/d

%o for i1 in range(E1 + 1):

%o     for i2 in range(E2 + 1):

%o         p[0][i1][i2] = 1

%o for n in range(1, E1 + E2):

%o     for s in range(n+1):

%o         for k1 in range(E1+1):

%o             for k2 in range(E2+1):

%o                 for i1 in range(k1 + 1):

%o                     for i2 in range(k2 + 1):

%o                         p[n][k1][k2] += 1/n * s * q[s][k1-i1][k2-i2] * p[n-s][i1][i2]

%o A = [[ p[n1+n2-1][n1][n2] for n1 in range(E1+1)] for n2 in range(E2+1)]

%Y Cf. A003262, which is the univariate variant of this sequence.

%Y Cf. A172003, which is the analogous sequence for implicit divided differences, and A162326 for its univariate variant.

%K nonn,tabl

%O 1,3

%A _Georg Muntingh_, Jan 22 2010