cp's OEIS Frontend

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.

%I A071920 #24 Dec 20 2020 08:12:57
%S A071920 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 A071920 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 A071920 64,252,581,791,610,239,37,1,0
%N 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.
%C A071920 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 A071920 Alois P. Heinz, <a href="/A071920/b071920.txt">Rows n = 0..140, flattened</a>
%F A071920 a(n,m) = Sum_{k=0..m-1} binomial(n+2k-1, 2k) if n>0.
%e A071920 Square array a(n,m) begins:
%e A071920   0,    0,    0,    0,    0,    0,     0,     0,      0, ...
%e A071920   0,    1,    2,    3,    4,    5,     6,     7,      8, ...
%e A071920   0,    1,    4,    9,   16,   25,    36,    49,     64, ...
%e A071920   0,    1,    7,   22,   50,   95,   161,   252,    372, ...
%e A071920   0,    1,   11,   46,  130,  295,   581,  1036,   1716, ...
%e A071920   0,    1,   16,   86,  296,  791,  1792,  3612,   6672, ...
%e A071920   0,    1,   22,  148,  610, 1897,  4900, 11088,  22716, ...
%e A071920   0,    1,   29,  239, 1163, 4166, 12174, 30738,  69498, ...
%e A071920   0,    1,   37,  367, 2083, 8518, 27966, 78354, 194634, ...
%p A071920 a:= (n, m)-> `if`(n=0, 0, add(binomial(n+2*j-1, 2*j), j=0..m-1)):
%p A071920 seq(seq(a(n, d-n), n=0..d), d=0..10);  # _Alois P. Heinz_, Sep 21 2013
%t A071920 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 A071920 Cf. A071921, A225010.
%Y A071920 Main diagonal is A088536.
%K A071920 nonn,easy,tabl
%O A071920 0,8
%A A071920 Michele Dondi (bik.mido(AT)tiscalinet.it), Jun 14 2002