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%I #11 Mar 08 2020 00:04:52
%S 1,1,3,5,3,13,33,33,13,71,245,351,245,71,441,1921,3597,3597,1921,441,
%T 2955,15525,35931,46709,35931,15525,2955,20805,127905,352665,563821,
%U 563821,352665,127905,20805,151695,1067925,3417975,6483285,7963151
%N Let y = y(u,v) be implicitly defined by g(u,v,y(u,v)) = 0. Read as a triangle by rows, the sequence represents the number of terms a(i,k-i) in the expansion of the bivariate divided difference [u_0,...,u_i; v_0,...,v_{k-i}]y in terms of trivariate divided differences 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),...
%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} of the generating function
%F - log(1 - Sum_{(r,s,t) in E} u^r * v^s * y^{r+s+t-1})
%F = Sum_{q >= 1} (Sum_{(r,s,t) in E} u^r * v^s * y^{r+s+t-1})^q / q.
%e 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 A162326.
%e For (m,n) = (1,1), one expresses [u_0,u_1;v_0,v_1]y as a sum of 5 terms,
%e [01;01]y =
%e - [0;0;(0,0),(1,0),(1,1)]g * [01;0;(1,0)]g * [1;01;(1,1)]g /
%e ( [0;0;(0,0),(1,1)]g * [0;0;(0,0),(1,0)]g * [1;0;(1,0),(1,1)]g )
%e + [01;0;(1,0),(1,1)]g * [1;01;(1,1)]g /
%e ( [0;0;(0,0),(1,1)]g * [1;0;(1,0),(1,1)]g )
%e - [01;01;(1,1)]g / [0;0;(0,0),(1,1)]g
%e - [0;0;(0,0),(0,1),(1,1)]g * [0;01;(0,1)]g * [01;1;(1,1)]g /
%e ( [0;0;(0,0),(1,1)]g * [0;0;(0,0),(0,1)]g * [0;1;(0,1),(1,1)]g )
%e + [0;01;(0,1),(1,1)]g * [01;1;(1,1)]g /
%e ( [0;0;(0,0),(1,1)]g * [0;1;(0,1),(1,1)]g ),
%e where the numbers refer to the indices of the corresponding variable, e.g.
%e [1;01;(1,1)]g = [u_1;v_0,v_1;y(u_1,v_1)]g.
%o (Sage)
%o R.<X1,X2,Y> = PolynomialRing(ZZ,3)
%o def P(n1,n2,q):
%o E = cartesian_product([list(range(n1+1)), list(range(n2+1)), list(range(n1+n2+1))])
%o E = [(i1,i2,j) for (i1,i2,j) in E if (i1,i2,j) != (0,0,0) and
%o (i1,i2,j) != (0,0,1) and i1 + i2 + j <= n1 + n2 and
%o 2*(i1 + i2) + j - 1 <= 2*(n1+n2) - q]
%o return R.sum(X1^s1 * X2^s2 * Y^(s1+s2+t-1) for s1,s2,t in E)
%o n1, n2 = 4, 4
%o L = [[0 for _ in range(n1 + 1)]] * (n2 + 1)
%o h = 1 + sum(((P(n1,n2,q))^q)/q for q in range(1,2*(n1+n2)))
%o for k1 in range(n1+1):
%o for k2 in range(k1+1):
%o if (k1, k2) != (0, 0):
%o print(k1, k2, h.coefficient({X1:k1, X2:k2, Y:k1+k2-1}))
%Y Cf. A162326, which is the univariate variant of this sequence.
%Y Cf. A172004, which is the analogous sequence for implicit derivatives, and A003262 for its univariate variant.
%K nonn,tabl
%O 1,3
%A _Georg Muntingh_, Jan 22 2010
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