OFFSET
1,2
COMMENTS
T(n,k) is the number of compositions [p(1), p(2), ..., p(k)] of n such that max(p(j) - p(j-1)) == k.
The first column is A000041 (partition numbers).
Sum of first and second column is A003116.
Sum of the first three columns is A224959.
The second columns deviates from A054454 after the term 600.
Row sums are A011782.
LINKS
Joerg Arndt and Alois P. Heinz, Rows n = 1..141, flattened
EXAMPLE
Triangle starts:
01: 1,
02: 2, 0,
03: 3, 1, 0,
04: 5, 2, 1, 0,
05: 7, 6, 2, 1, 0,
06: 11, 12, 6, 2, 1, 0,
07: 15, 26, 14, 6, 2, 1, 0,
08: 22, 50, 33, 14, 6, 2, 1, 0,
09: 30, 97, 72, 34, 14, 6, 2, 1, 0,
10: 42, 180, 156, 77, 34, 14, 6, 2, 1, 0,
11: 56, 332, 328, 173, 78, 34, 14, 6, 2, 1, 0,
12: 77, 600, 681, 378, 177, 78, 34, 14, 6, 2, 1, 0,
13: 101, 1078, 1393, 818, 393, 178, 78, 34, 14, 6, 2, 1, 0,
14: 135, 1917, 2821, 1746, 863, 397, 178, 78, 34, 14, 6, 2, 1, 0,
15: 176, 3393, 5660, 3695, 1872, 877, 398, 178, 78, 34, 14, 6, 2, 1, 0,
...
The fifth row corresponds to the following statistics:
#: M composition
01: 0 [ 1 1 1 1 1 ]
02: 1 [ 1 1 1 2 ]
03: 1 [ 1 1 2 1 ]
04: 2 [ 1 1 3 ]
05: 1 [ 1 2 1 1 ]
06: 1 [ 1 2 2 ]
07: 2 [ 1 3 1 ]
08: 3 [ 1 4 ]
09: 0 [ 2 1 1 1 ]
10: 1 [ 2 1 2 ]
11: 0 [ 2 2 1 ]
12: 1 [ 2 3 ]
13: 0 [ 3 1 1 ]
14: 0 [ 3 2 ]
15: 0 [ 4 1 ]
16: 0 [ 5 ]
There are 7 compositions with no up-step (M=0), 6 with M=1, 2 with M=2, and 1 with M=3.
MAPLE
b:= proc(n, v) option remember; `if`(n=0, 1, add((p->
`if`(i<v, add(coeff(p, x, h)*x^`if`(h<v-i, v-i, h),
h=0..degree(p)), p))(b(n-i, i)), i=1..n))
end:
T:= n-> seq(coeff(b(n, 0), x, i), i=0..n-1):
seq(T(n), n=1..14); # Alois P. Heinz, Feb 22 2014
MATHEMATICA
b[n_, v_] := b[n, v] = If[n == 0, 1, Sum[Function[{p}, If[i<v, Sum[Coefficient[p, x, h]*x^If[h<v-i, v-i, h], {h, 0, Exponent[p, x]}], p]][b[n-i, i]], {i, 1, n}]] ; T[n_] := Table[Coefficient[b[n, 0], x, i], {i, 0, n-1}]; Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Joerg Arndt, Apr 27 2013
STATUS
approved