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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
OFFSET
1,3
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.
T(n, m) = A001263(n, m)*hypergeom([1 - m, 2], [n - m + 2], -1). - Peter Luschny, May 06 2021
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
Cf. A001263.
Sequence in context: A274105 A366156 A056242 * A128718 A112358 A126351
KEYWORD
nonn,tabl
AUTHOR
Yuriy Shablya, May 05 2021
STATUS
approved