G. C. Greubel, <a href="/A135021/b135021_1.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

reviewed

approved

proposed

reviewed

editing

proposed

From Bruno Berselli, Dec 08 2012: (Start)

1 , 1;

[Bruno Berselli, Dec 08 2012]

(End)

proposed

editing

editing

proposed

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 16, 6, 1, 1, 1, 125, 70, 10, 1, 1, 1, 1296, 1215, 200, 15, 1, 1, 1, 16807, 27951, 5915, 455, 21, 1, 1, 1, 262144, 799708, 229376, 20230, 896, 28, 1, 1, 1, 4782969, 27337500, 10946964, 1166886, 55566, 1596, 36, 1, 1, 1, 100000000, 1086190605, 618435840, 82031250, 4429152, 131250, 2640, 45, 1, 1, 1, 2357947691, 49162945645, 40283203125, 6768679170, 426666702, 13763442, 277530, 4125, 55, 1, 1

Triangle read by rows: T(n,r) = number of maximum r-uniform acyclic hypergraphs of order n and size n-r+1, 1 <= r <= n+1.

G. C. Greubel, <a href="/A135021/b135021_1.txt">Table of n, a(n) for the first 50 n = 0..1325</a> (rows</a> 0..50)

1,;

1, 1;

1, 3, 1, 1;

1, 16, 6, 3, 1, 1;

1, 125, 70, 10, 16, 6, 1, 1;

1, 1296, 1215, 200, 15, 125, 70, 10, 1, 1;

1, 16807, 27951, 5915, 455, 21, 1296, 1215, 200, 15, 1, 1;

1, 262144, 799708, 229376, 20230, 896, 28, 16807, 27951, 5915, 455, 21, 1, 1;

1, 262144, 799708, 229376, 20230, 896, 28, 1, 1;

1, 4782969, 27337500, 10946964, 1166886, 55566, 1596, 36, 1, etc.1;

Cf. A370770 (unlabeled version).

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 16, 6, 1, 1, 1, 125, 70, 10, 1, 1, 1, 1296, 1215, 200, 15, 1, 1, 1, 16807, 27951, 5915, 455, 21, 1, 1, 1, 262144, 799708, 229376, 20230, 896, 28, 1, 1, 1, 4782969, 27337500, 10946964, 1166886, 55566, 1596, 36, 1, 1, 1, 100000000, 1086190605, 618435840, 82031250, 4429152, 131250, 2640, 45, 1, 1, 1, 2357947691, 49162945645, 40283203125, 6768679170, 426666702, 13763442, 277530, 4125, 55, 1, 1

1,5

0,8

T(n,r) is the number of (r-1)-trees on n nodes. - Andrew Howroyd, Mar 02 2024

T(n,r) = Cbinomial(n,r-1)*(n*(r-1)-r^2+2r2*r)^(n-r-1).

(PARI) T(n, r) = binomial(n, r-1)*(n*(r-1)-r^2+2*r)^(n-r-1) \\ Andrew Howroyd, Mar 02 2024

Columns 1..5 are A000012, A000272, A036361, A036362, A036506.

Diagonal r=n+1 inserted by Andrew Howroyd, Mar 02 2024

1,

1, 1;

1, 3, 1;

1, 16, 6, 1;

1, 125, 70, 10, 1;

1, 1296, 1215, 200, 15, 1;

1, 16807, 27951, 5915, 455, 21, 1;

1, 262144, 799708, 229376, 20230, 896, 28, 1;

1, 4782969, 27337500, 10946964, 1166886, 55566, 1596, 36, 1, etc.

Columns A000012, A000272, A036361, A036362, A036506.

approved

editing