OFFSET
0,3
COMMENTS
In the following description the alphabet {0..r} is taken as a basis, with r = 3 in this case.
For example, the quaternary word 2|03|123|3 of length n=7, has m=4 strictly increasing runs.
The empty word has n = 0 and m = 0, and T(0, 0) = 1.
T(n, 0) = 0 for n >= 1.
T(n, m) <> 0 for m <= n <= m*(r+1). T(m*(r+1), m) = 1.
T(n,m) is a partition, based on m, of all the words of length n, so Sum_{k=0..n} T(n,k) = (r+1)^n.
REFERENCES
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, p. 24, p. 154.
LINKS
FORMULA
Refer to comment to A120987 concerning formulas for general values of r and considerations.
Therefrom we get
T(n, m) = Qsc(3, n, m) =
Nb(4*m-n, 3, n+1) = Nb(4*(n-m)+3, 3, n+1) =
Sum_{j=0..n+1} (-1)^j*Cb(n+1, j)*Cb(4*(m-j), 4*(m-j)-n) =
Sum_{j=0..m} (-1)^(m-j)*Cb(n+1, m-j)*Cb(4*j, n) =
(in this last version Cb(n,m) can be replaced by binomial(n,m))
Sum_{j=0..m} (-1)^(m-j)*binomial(n+1, m-j)*binomial(4*j, n) = [z^n, t^m](1-t)/(1-t(1+(1-t)z)^4) where [x^n]F(x) denotes the coefficient of x^n in the formal power series expansion of F(x),
Nb(s,r,n) denotes the (r+1)-nomial coefficient [x^s](1+x+..+x^r)^n,(Nb(s,3,n) = A008287(n,s)).
Cb(x,m) denotes the binomial coefficient in its extended falling factorial notation (Cb(x,m)= x^_m/m! iff m is a nonnegative integer, 0 otherwise), as defined in the Graham et al. reference.
The diagonal T(n, n) = Nb(3, 3, n+1) = Sum_{j=0..n} (-1)^(n-j)*Cb(n+1, n-j)*Cb(4*j, n) = Cb(n+3, 3) = binomial(n+3, 3) = A000292(n+1).
EXAMPLE
Triangle starts:
1;
0, 4;
0, 6, 10;
0, 4, 40, 20;
0, 1, 65, 155, 35;
0, 0, 56, 456, 456, 56;
.
T(3,2) = 40, which accounts for the following words:
[0 <= a <= 0, 1 | 0 <= b <= 1] = 2
[0 <= a <= 1, 2 | 0 <= b <= 2] = 6
[0 <= a <= 2, 3 | 0 <= b <= 3] = 12
[0 <= a <= 3 | 0, 1 <= b <= 3] = 12
[1 <= a <= 3 | 1, 2 <= b <= 3] = 6
[2 <= a <= 3 | 2, 3 <= b <= 3] = 2
PROG
(MuPAD) T:=(n, m)->_plus((-1)^(m-j)*binomial(n+1, m-j)*binomial(4*j, n)$j=0..m):
(PARI) T(n, k) = sum(j=0, k, (-1)^(k-j)*binomial(n+1, k-j)*binomial(4*j, n));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print()); \\ Michel Marcus, Feb 09 2016
CROSSREFS
KEYWORD
tabl,nonn
AUTHOR
Giuliano Cabrele, Dec 13 2015
STATUS
approved