|
|
A343960
|
|
Triangle read by rows: T(n,m) = Sum_{k=1..m} (k/n)*binomial(n,m-k)*binomial(n,m), n >= m >= 1.
|
|
0
|
|
|
1, 1, 2, 1, 5, 4, 1, 9, 17, 8, 1, 14, 46, 49, 16, 1, 20, 100, 180, 129, 32, 1, 27, 190, 510, 603, 321, 64, 1, 35, 329, 1225, 2121, 1827, 769, 128, 1, 44, 532, 2618, 6202, 7700, 5164, 1793, 256, 1, 54, 816, 5124, 15876, 26628, 25392, 13878, 4097, 512
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
LINKS
|
|
|
FORMULA
|
T(n,m) = Sum_{k=1..m} (k/n)*binomial(n,m-k)*binomial(n,m).
G.f.: N(x,y)/(1-N(x,y)), where N(x,y) is a g.f. for the Narayana numbers A001263.
|
|
EXAMPLE
|
Triangle begins:
---------------------------------------------------------------------
n \ m | 1 2 3 4 5 6 7 8 9 10
-------+-------------------------------------------------------------
1 | 1
2 | 1 2
3 | 1 5 4
4 | 1 9 17 8
5 | 1 14 46 49 16
6 | 1 20 100 180 129 32
7 | 1 27 190 510 603 321 64
8 | 1 35 329 1225 2121 1827 769 128
9 | 1 44 532 2618 6202 7700 5164 1793 256
10 | 1 54 816 5124 15876 26628 25392 13878 4097 512
|
|
MATHEMATICA
|
T[n_, m_] := Sum[Binomial[n, m - k] * Binomial[n, m] * k/n, {k, 1, n}]; Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Amiram Eldar, May 06 2021 *)
|
|
PROG
|
(Maxima)
T(n, m):=sum((k/n)*binomial(n, m-k)*binomial(n, m), k, 1, m)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|