OFFSET
1,5
COMMENTS
Taking finite differences of array columns from the top down, we obtain (1; 1,1; 1,2,1; 1,4,2,1; ...) = A048004 rows. - Gary W. Adamson, Aug 20 2010
T(m,n) is the number of binary words of length n-1 with < m consecutive 1's. - Geoffrey Critzer, Sep 02 2012
REFERENCES
J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, Princeton, NJ, 1978, p. 154.
LINKS
Alois P. Heinz, Antidiagonals n = 1..141, flattened
Hsin-Po Wang and Chi-Wei Chin, On Counting Subsequences and Higher-Order Fibonacci Numbers, arXiv:2405.17499 [cs.IT], 2024. See p. 2.
FORMULA
G.f.: (1-z)/[1-2z+z^(t+1)].
EXAMPLE
T(2,5) counts 11111, 1112, 1121, 1211, 2111, 122, 212, 221, where "1211" abbreviates the composition 1+2+1+1.
These eight compositions correspond respectively to: {0,0,0,0}, {0,0,0,1}, {0,0,1,0}, {0,1,0,0}, {1,0,0,0}, {0,1,0,1}, {1,0,0,1}, {1,0,1,0} per the bijection given by N. J. A. Sloane in A048004. - Geoffrey Critzer, Sep 02 2012
The array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 5, 8, 13, ...
1, 2, 4, 7, 13, ...
1, 2, 4, 8, ...
1, 2, 4, ...
1, 2, ...
1, ...
MAPLE
G := t->(1-z)/(1-2*z+z^(t+1)): T := (m, n)->coeff(series(G(m), z=0, 30), z^n): matrix(7, 12, T);
# second Maple program:
T:= proc(m, n) option remember; `if`(n=0 or m=1, 1,
add(T(m, n-j), j=1..min(n, m)))
end:
seq(seq(T(1+d-n, n), n=1..d), d=1..14); # Alois P. Heinz, May 21 2013
MATHEMATICA
Table[nn=10; a=(1-x^k)/(1-x); b=1/(1-x); c=(1-x^(k-1))/(1-x); CoefficientList[ Series[a b/(1-x^2 b c), {x, 0, nn}], x], {k, 1, nn}]//Grid (* Geoffrey Critzer, Sep 02 2012 *)
T[m_, n_] := T[m, n] = If[n == 0 || m == 1, 1, Sum[T[m, n-j], {j, 1, Min[n, m]}]]; Table[Table[T[1+d-n, n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Nov 12 2014, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved