A094953 - OEIS

%I #9 Dec 03 2018 05:06:32

%S 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,

%T 4,32,102,184,200,132,49,8,5,40,150,320,430,372,203,64,9,5,50,210,520,

%U 830,888,637,296,81,10,6,60,285,800,1480,1884,1673,1024,414,100,11,6

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

%H S. Heubach and T. Mansour, <a href="https://arxiv.org/abs/math/0310197">Counting rises, levels and drops in compositions</a>, arXiv:math/0310197 [math.CO], 2003.

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

%e 1

%e 1 2

%e 2 4 3

%e 2 8 9 4

%e 3 12 21 16 5

%e 3 18 39 44 25 6

%e 4 24 66 96 80 36 7

%t T[n_, m_] := SeriesCoefficient[(m-1)x^(m+1)/(1+x)/(1-x)^m, {x, 0, n+1}];

%t Table[T[n, m], {n, 2, 13}, {m, 2, n}] // Flatten (* _Jean-François Alcover_, Dec 03 2018 *)

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

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

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

%Y Cf. A045927.

%K nonn,tabl

%O 2,3

%A _Ralf Stephan_, May 26 2004