OFFSET
0,8
COMMENTS
If one uses a definition of unimodality that involves universal quantifiers on the domain of a function then a(0,m)=1 a priori.
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
Kenneth Edwards and Michael A. Allen, New Combinatorial Interpretations of the Fibonacci Numbers Squared, Golden Rectangle Numbers, and Jacobsthal Numbers Using Two Types of Tile, arXiv:2009.04649 [math.CO], 2020.
FORMULA
a(n,m) = 1 if n=0, m>=0, a(n,m) = Sum_{k=0..m-1} C(2k+n-1,2k) otherwise.
EXAMPLE
Square array a(n,m) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, 7, 8, ...
0, 1, 4, 9, 16, 25, 36, 49, 64, ...
0, 1, 7, 22, 50, 95, 161, 252, 372, ...
0, 1, 11, 46, 130, 295, 581, 1036, 1716, ...
0, 1, 16, 86, 296, 791, 1792, 3612, 6672, ...
0, 1, 22, 148, 610, 1897, 4900, 11088, 22716, ...
0, 1, 29, 239, 1163, 4166, 12174, 30738, 69498, ...
0, 1, 37, 367, 2083, 8518, 27966, 78354, 194634, ...
MAPLE
a:= (n, m)-> `if`(n=0, 1, add(binomial(n+2*j-1, 2*j), j=0..m-1)):
seq(seq(a(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Sep 22 2013
MATHEMATICA
a[0, 0] = 1; a[n_, m_] := Sum[Binomial[2k+n-1, 2k], {k, 0, m-1}]; Table[a[n - m, m], {n, 0, 12}, {m, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 11 2015 *)
CROSSREFS
KEYWORD
AUTHOR
Michele Dondi (bik.mido(AT)tiscalinet.it), Jun 14 2002
STATUS
approved