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A225084 Triangle read by rows: T(n,k) is the number of compositions of n with maximal up-step k; n>=1, 0<=k<n. 5
1, 2, 0, 3, 1, 0, 5, 2, 1, 0, 7, 6, 2, 1, 0, 11, 12, 6, 2, 1, 0, 15, 26, 14, 6, 2, 1, 0, 22, 50, 33, 14, 6, 2, 1, 0, 30, 97, 72, 34, 14, 6, 2, 1, 0, 42, 180, 156, 77, 34, 14, 6, 2, 1, 0, 56, 332, 328, 173, 78, 34, 14, 6, 2, 1, 0, 77, 600, 681, 378, 177, 78, 34, 14, 6, 2, 1, 0, 101, 1078, 1393, 818, 393, 178, 78, 34, 14, 6, 2, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
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
Cf. A225085 (partial sums of rows).
T(2n,n) gives A059570.
Sequence in context: A225624 A168020 A321878 * A238345 A299070 A209599
KEYWORD
nonn,tabl
AUTHOR
Joerg Arndt, Apr 27 2013
STATUS
approved

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Last modified March 19 02:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)