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 A172003 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. 3
 1, 1, 3, 5, 3, 13, 33, 33, 13, 71, 245, 351, 245, 71, 441, 1921, 3597, 3597, 1921, 441, 2955, 15525, 35931, 46709, 35931, 15525, 2955, 20805, 127905, 352665, 563821, 563821, 352665, 127905, 20805, 151695, 1067925, 3417975, 6483285, 7963151 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The sequence starts with a(1,0),a(0,1),a(2,0),a(1,1),a(0,2),a(3,0),... LINKS Table of n, a(n) for n=1..40. FORMULA 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 - log(1 - Sum_{(r,s,t) in E} u^r * v^s * y^{r+s+t-1}) = Sum_{q >= 1} (Sum_{(r,s,t) in E} u^r * v^s * y^{r+s+t-1})^q / q. EXAMPLE 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. For (m,n) = (1,1), one expresses [u_0,u_1;v_0,v_1]y as a sum of 5 terms, [01;01]y = - [0;0;(0,0),(1,0),(1,1)]g * [01;0;(1,0)]g * [1;01;(1,1)]g / ( [0;0;(0,0),(1,1)]g * [0;0;(0,0),(1,0)]g * [1;0;(1,0),(1,1)]g ) + [01;0;(1,0),(1,1)]g * [1;01;(1,1)]g / ( [0;0;(0,0),(1,1)]g * [1;0;(1,0),(1,1)]g ) - [01;01;(1,1)]g / [0;0;(0,0),(1,1)]g - [0;0;(0,0),(0,1),(1,1)]g * [0;01;(0,1)]g * [01;1;(1,1)]g / ( [0;0;(0,0),(1,1)]g * [0;0;(0,0),(0,1)]g * [0;1;(0,1),(1,1)]g ) + [0;01;(0,1),(1,1)]g * [01;1;(1,1)]g / ( [0;0;(0,0),(1,1)]g * [0;1;(0,1),(1,1)]g ), where the numbers refer to the indices of the corresponding variable, e.g. [1;01;(1,1)]g = [u_1;v_0,v_1;y(u_1,v_1)]g. PROG (Sage) R. = PolynomialRing(ZZ, 3) def P(n1, n2, q): E = cartesian_product([list(range(n1+1)), list(range(n2+1)), list(range(n1+n2+1))]) E = [(i1, i2, j) for (i1, i2, j) in E if (i1, i2, j) != (0, 0, 0) and (i1, i2, j) != (0, 0, 1) and i1 + i2 + j <= n1 + n2 and 2*(i1 + i2) + j - 1 <= 2*(n1+n2) - q] return R.sum(X1^s1 * X2^s2 * Y^(s1+s2+t-1) for s1, s2, t in E) n1, n2 = 4, 4 L = [[0 for _ in range(n1 + 1)]] * (n2 + 1) h = 1 + sum(((P(n1, n2, q))^q)/q for q in range(1, 2*(n1+n2))) for k1 in range(n1+1): for k2 in range(k1+1): if (k1, k2) != (0, 0): print(k1, k2, h.coefficient({X1:k1, X2:k2, Y:k1+k2-1})) CROSSREFS Cf. A162326, which is the univariate variant of this sequence. Cf. A172004, which is the analogous sequence for implicit derivatives, and A003262 for its univariate variant. Sequence in context: A266603 A121278 A023587 * A244801 A002586 A258811 Adjacent sequences: A172000 A172001 A172002 * A172004 A172005 A172006 KEYWORD nonn,tabl AUTHOR Georg Muntingh, Jan 22 2010 STATUS approved

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Last modified May 29 01:48 EDT 2023. Contains 363029 sequences. (Running on oeis4.)