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

0,8

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.

Links

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

Formula

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

Example

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, ...

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 *)

Crossrefs

Cf. A071921, A225010.

Main diagonal is A088536.

Sequence in context: A099423 A221515 A221984 * A306548 A320531 A345698

Adjacent sequences: A071917 A071918 A071919 * A071921 A071922 A071923

Keyword

nonn,easy,tabl

Author

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

Status

approved