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

0,8

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.

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.

Formula

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

Example

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, ...

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 *)

Crossrefs

Cf. A071920, A225010.

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

Sequence in context: A118340 A213276 A210391 * A003992 A337161 A246118

Adjacent sequences: A071918 A071919 A071920 * A071922 A071923 A071924

Keyword

nonn,easy,tabl

Author

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

Status

approved