A094953 Triangle T(n,m) read by rows: number of rises (drops) in the compositions of n with m parts, m>=2.

1, 1, 2, 2, 4, 3, 2, 8, 9, 4, 3, 12, 21, 16, 5, 3, 18, 39, 44, 25, 6, 4, 24, 66, 96, 80, 36, 7, 4, 32, 102, 184, 200, 132, 49, 8, 5, 40, 150, 320, 430, 372, 203, 64, 9, 5, 50, 210, 520, 830, 888, 637, 296, 81, 10, 6, 60, 285, 800, 1480, 1884, 1673, 1024, 414, 100, 11, 6

Offset: 2

Author

Ralf Stephan, May 26 2004

Keywords

• nonn
• tabl

Examples

			1
1 2
2 4 3
2 8 9 4
3 12 21 16 5
3 18 39 44 25 6
4 24 66 96 80 36 7
		

Links

• S. Heubach and T. Mansour, Counting rises, levels and drops in compositions, arXiv:math/0310197 [math.CO], 2003.

Crossrefs

Columns 2-4 (+-offset) are A004526, A007590, A007518.

Row sums are A045883, diagonals include n, n^2, (n-1)(n^2-n+2)/2, (n-1)^2(n^+n+6), etc.

Cf. A045927.

Programs

Mathematica

T[n_, m_] := SeriesCoefficient[(m-1)x^(m+1)/(1+x)/(1-x)^m, {x, 0, n+1}];
Table[T[n, m], {n, 2, 13}, {m, 2, n}] // Flatten (* Jean-François Alcover, Dec 03 2018 *)

PARI

T(n,m)=polcoeff((m-1)*x^(m+1)/(1+x)/(1-x)^m,n)

Formula

G.f. of m-th column: [(m-1)x^(m+1)]/[(1+x)(1-x)^m].