A071921 Square array giving number of unimodal functions [n]->[m] for n>=0, m>=0, with a(0,m)=1 by definition, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 7, 1, 0, 1, 5, 16, 22, 11, 1, 0, 1, 6, 25, 50, 46, 16, 1, 0, 1, 7, 36, 95, 130, 86, 22, 1, 0, 1, 8, 49, 161, 295, 296, 148, 29, 1, 0, 1, 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 universal quantifiers on the domain of a function then a(0,m)=1 a priori.

Examples

			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, ...
		

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.

Crossrefs

Cf. A071920, A225010.

Main diagonal gives A088536 (for n>=1).

Programs

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 *)

Formula

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