%I #24 Dec 20 2020 08:12:57

%S 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,

%T 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,

%U 64,252,581,791,610,239,37,1,0

%N 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.

%C If one uses a definition of unimodality that involves existential quantifiers on the domain of a function then a(0,m)=0 a priori.

%H Alois P. Heinz, <a href="/A071920/b071920.txt">Rows n = 0..140, flattened</a>

%F a(n,m) = Sum_{k=0..m-1} binomial(n+2k-1, 2k) if n>0.

%e Square array a(n,m) begins:

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

%e 0, 1, 2, 3, 4, 5, 6, 7, 8, ...

%e 0, 1, 4, 9, 16, 25, 36, 49, 64, ...

%e 0, 1, 7, 22, 50, 95, 161, 252, 372, ...

%e 0, 1, 11, 46, 130, 295, 581, 1036, 1716, ...

%e 0, 1, 16, 86, 296, 791, 1792, 3612, 6672, ...

%e 0, 1, 22, 148, 610, 1897, 4900, 11088, 22716, ...

%e 0, 1, 29, 239, 1163, 4166, 12174, 30738, 69498, ...

%e 0, 1, 37, 367, 2083, 8518, 27966, 78354, 194634, ...

%p a:= (n, m)-> `if`(n=0, 0, add(binomial(n+2*j-1, 2*j), j=0..m-1)):

%p seq(seq(a(n, d-n), n=0..d), d=0..10); # _Alois P. Heinz_, Sep 21 2013

%t 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 *)

%Y Cf. A071921, A225010.

%Y Main diagonal is A088536.

%K nonn,easy,tabl

%O 0,8

%A Michele Dondi (bik.mido(AT)tiscalinet.it), Jun 14 2002