login
A191588
T(m,n) is the number of ways to split two strings of length m and n, respectively, into the same number of nonempty parts such that at least one of the corresponding parts has length 1 and such that the parts have at most size 2.
0
1, 1, 1, 0, 2, 3, 0, 1, 3, 7, 0, 0, 3, 7, 13, 0, 0, 1, 6, 17, 27, 0, 0, 0, 4, 14, 36, 61, 0, 0, 0, 1, 10, 35, 77, 133, 0, 0, 0, 0, 5, 25, 81, 173, 287, 0, 0, 0, 0, 1, 15, 65, 183, 387, 633, 0, 0, 0, 0, 0, 6, 41, 161, 421, 857, 1407, 0, 0, 0, 0, 0, 1, 21, 112, 385, 969, 1911, 3121, 0, 0, 0, 0, 0, 0, 7, 63, 294, 918, 2211, 4287, 6943, 0, 0, 0, 0, 0, 0, 1, 28, 182, 742, 2181, 5040, 9619, 15517, 0, 0
OFFSET
1,5
COMMENTS
Diagonal appears to be A098479. - Joerg Arndt, Jun 09 2011
T(m,n) is the number of lattice paths from (0,0) to (m,n) with steps in {(1,1),(1,2),(2,1)}. - Steffen Eger, Sep 25 2012
LINKS
FORMULA
For m >= n: T(m,n) = C(n,2*n-m) + Sum_{k=2..n-1} C(k,2*k-n)*C(2*k-n,3*k-n-m) (note: C(2*k-n,3*k-n-m) = C(2*k-n,m-k)) where C(n,k) = binomial(n,k) for n >= k and 0 otherwise.
Symmetrically extended by T(n,m) = T(m,n).
EXAMPLE
1
1 1
0 2 3
0 1 3 7
0 0 3 7 13
0 0 1 6 17 27
0 0 0 4 14 36 61
0 0 0 1 10 35 77 133
0 0 0 0 5 25 81 173 287
0 0 0 0 1 15 65 183 387 633
0 0 0 0 0 6 41 161 421 857 1407
0 0 0 0 0 1 21 112 385 969 1911 3121
0 0 0 0 0 0 7 63 294 918 2211 4287 6943
0 0 0 0 0 0 1 28 182 742 2181 5040 9619 15517
0 0 0 0 0 0 0 8 92 504 1842 5134 11508 21602 34755
Examples:
For m=3, n=2, we have
x xx xx x
y y y y
For m=3, n=3, we have
x xx xx x x x x
yy y y yy y y y
For m=4, n=4, we have
x xx x x xx x xx x x xx x x x x xx x x xx x x x x
yy y y y y yy y yy y y y yy y yy y yy y y y y y y
MATHEMATICA
t[m_, n_] /; m >= n := t[m, n] = Binomial[n, 2n - m] + Sum[Binomial[k, 2k - n]*Binomial[2k - n, 3k - n - m], {k, 2, n-1}]; t[m_, n_] /; m < n := t[m, n]; Table[t[n, m], {n, 1, 14}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jan 07 2013, from formula *)
CROSSREFS
Cf. A180091, A185287, A098479 (diagonal).
Sequence in context: A112168 A072516 A320782 * A106450 A255961 A297328
KEYWORD
nonn,nice,tabl
AUTHOR
Steffen Eger, Jun 09 2011
STATUS
approved