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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. 4
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 (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-Fran├žois Alcover, Feb 18 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A225085 (partial sums of rows).

Sequence in context: A144955 A225624 A168020 * A238345 A209599 A238347

Adjacent sequences:  A225081 A225082 A225083 * A225085 A225086 A225087

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt, Apr 27 2013

STATUS

approved

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Last modified April 24 00:59 EDT 2017. Contains 285338 sequences.