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.

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

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.<X1, X2, Y> = 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