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),...
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
KEYWORD
nonn,tabl
AUTHOR
Georg Muntingh, Jan 22 2010
STATUS
approved