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.

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

Offset: 0

Author

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

Keywords

• nonn
• easy
• tabl

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.

Examples

			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, ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Cf. A071921, A225010.

Main diagonal is A088536.

Programs

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 *)

Formula

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