A007518 -id:A007518 - cp's OEIS frontend

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

Ralf Stephan, May 26 2004

			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

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

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.

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 *)

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

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

A248345 Signed version of A094953.

Original entry on oeis.org

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

Offset: 0

Derek Orr, Oct 30 2014

This is the transformation of the polynomial 1 + 2x + 3x^2 + 4x^3 + ... + n*x^(n-1)+(n+1)*x^n to the polynomial A_0*(x+1)^0 + A_1*(x+1)^1 + A_2*(x+1)^2 + ... + A_n*(x+1)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.

			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

Cf. A007518, A007590, A001057, A094953.

T(n,k)=(k+1)*sum(i=0,n-k,(-1)^i*binomial(i+k+1,k+1))
for(n=0,15,for(k=0,n,print1(T(n,k),", ")))

Rows sum to 1.
T(n,n) = n for n >= 0.
T(n,n-1) = -n^2 for n >= 1.
T(n,2) = A007518(n)*(-1)^n for n >= 2.
T(n,1) = A007590(n+1)*(-1)^(n+1) for n >= 1.
T(n,0) = A001057(n+1) for n >= 0.

cp's OEIS Frontend

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

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