The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A094953 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing all changes.

A094953

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

#9 by Michel Marcus at Mon Dec 03 05:06:32 EST 2018

STATUS

reviewed

approved

#8 by Joerg Arndt at Mon Dec 03 03:58:05 EST 2018

STATUS

proposed

reviewed

#7 by Michel Marcus at Mon Dec 03 01:50:21 EST 2018

STATUS

editing

proposed

#6 by Michel Marcus at Mon Dec 03 01:50:17 EST 2018

LINKS

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

STATUS

proposed

editing

#5 by Jean-François Alcover at Mon Dec 03 01:44:40 EST 2018

STATUS

editing

proposed

#4 by Jean-François Alcover at Mon Dec 03 01:44:35 EST 2018

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 *)

STATUS

approved

editing

#3 by Charles R Greathouse IV at Wed Apr 30 01:30:10 EDT 2014

AUTHOR

_Ralf Stephan, _, May 26 2004

Discussion

Wed Apr 30

01:30

OEIS Server: https://oeis.org/edit/global/2170

#2 by N. J. A. Sloane at Fri Feb 27 03:00:00 EST 2009

LINKS

S. Heubach and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0310197">Counting rises, levels, and drops in compositions</a>

KEYWORD

nonn,tabl,new

#1 by N. J. A. Sloane at Sat Jun 12 03:00:00 EDT 2004

NAME

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

DATA

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

2,3

LINKS

S. Heubach and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0310197">Counting rises, levels, and drops in compositions</a>

FORMULA

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

EXAMPLE

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

PROG

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

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.

KEYWORD

nonn,tabl

AUTHOR

Ralf Stephan, May 26 2004

STATUS

approved