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A071920 Square array giving number of unimodal functions [n]->[m] for n>=0, m>=0, with a(0,m)=0 for all m>=0, read by antidiagonals. 14
0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 3, 4, 1, 0, 0, 4, 9, 7, 1, 0, 0, 5, 16, 22, 11, 1, 0, 0, 6, 25, 50, 46, 16, 1, 0, 0, 7, 36, 95, 130, 86, 22, 1, 0, 0, 8, 49, 161, 295, 296, 148, 29, 1, 0, 0, 9, 64, 252, 581, 791, 610, 239, 37, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
COMMENTS
If one uses a definition of unimodality that involves existential quantifiers on the domain of a function then a(0,m)=0 a priori.
LINKS
FORMULA
a(n,m) = Sum_{k=0..m-1} binomial(n+2k-1, 2k) if n>0.
EXAMPLE
Square array a(n,m) begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, ...
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, 0, add(binomial(n+2*j-1, 2*j), j=0..m-1)):
seq(seq(a(n, d-n), n=0..d), d=0..10); # Alois P. Heinz, Sep 21 2013
MATHEMATICA
a[n_, m_] := Sum[Binomial[n+2*k-1, 2*k], {k, 0, m-1}]; a[0, _] = 0; Table[a[n-m, m], {n, 0, 10}, {m, n, 0, -1}] // Flatten (* Jean-François Alcover, Feb 25 2015 *)
CROSSREFS
Main diagonal is A088536.
Sequence in context: A099423 A221515 A221984 * A306548 A320531 A345698
KEYWORD
nonn,easy,tabl
AUTHOR
Michele Dondi (bik.mido(AT)tiscalinet.it), Jun 14 2002
STATUS
approved

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Last modified April 23 06:04 EDT 2024. Contains 371906 sequences. (Running on oeis4.)