A088536 -id:A088536 - cp's OEIS frontend

A223718 Number of unimodal functions [1..n]->[0..2].

1, 3, 9, 22, 46, 86, 148, 239, 367, 541, 771, 1068, 1444, 1912, 2486, 3181, 4013, 4999, 6157, 7506, 9066, 10858, 12904, 15227, 17851, 20801, 24103, 27784, 31872, 36396, 41386, 46873, 52889, 59467, 66641, 74446, 82918, 92094, 102012, 112711, 124231

Offset: 0

R. H. Hardin, Mar 26 2013

Column 1 of A223725.

			Some solutions for n=3
..1....2....0....1....0....2....1....2....0....2....0....1....0....1....0....1
..2....1....1....1....0....0....0....1....0....2....2....1....1....2....2....2
..0....1....0....0....1....0....0....0....2....2....1....1....1....2....0....1
From _Joerg Arndt_, Dec 27 2023: (Start)
The a(3) = 22 such functions are (dots for zeros)
   1:  [ . . . ]
   2:  [ . . 1 ]
   3:  [ . . 2 ]
   4:  [ . 1 . ]
   5:  [ . 1 1 ]
   6:  [ . 1 2 ]
   7:  [ . 2 . ]
   8:  [ . 2 1 ]
   9:  [ . 2 2 ]
  10:  [ 1 . . ]
  11:  [ 1 1 . ]
  12:  [ 1 1 1 ]
  13:  [ 1 1 2 ]
  14:  [ 1 2 . ]
  15:  [ 1 2 1 ]
  16:  [ 1 2 2 ]
  17:  [ 2 . . ]
  18:  [ 2 1 . ]
  19:  [ 2 1 1 ]
  20:  [ 2 2 . ]
  21:  [ 2 2 1 ]
  22:  [ 2 2 2 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n=1..210 from R. H. Hardin)
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.

Column m=3 of A071920.

Cf. A000124 (unimodal functions [1..n]->[0..1]), A088536 ([1..n] -> [1..n]).

a(n) = A071920(n,3) = 1+n*(n+1)*(n^2+5*n+18)/24.

G.f.: 1-x*(x^2-3*x+3)*(x^2-x+1) / (x-1)^5 . a(n) = 1+A051744(n). - R. J. Mathar, May 17 2014

a(0)=1 prepended by Alois P. Heinz, Dec 27 2023

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.

Original entry on oeis.org

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

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

If one uses a definition of unimodality that involves existential quantifiers on the domain of a function then a(0,m)=0 a priori.

			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, ...

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

Cf. A071921, A225010.

Main diagonal is A088536.

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

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 *)

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

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.

Original entry on oeis.org

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

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

If one uses a definition of unimodality that involves universal quantifiers on the domain of a function then a(0,m)=1 a priori.

			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, ...

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.

Cf. A071920, A225010.

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

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

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 *)

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

A225006 Number of n X n 0..1 arrays with rows unimodal and columns nondecreasing.

Original entry on oeis.org

1, 2, 9, 50, 295, 1792, 11088, 69498, 439791, 2803658, 17978389, 115837592, 749321716, 4863369656, 31655226108, 206549749930, 1350638103791, 8848643946550, 58069093513635, 381650672631330, 2511733593767295, 16550500379912640, 109176697072162080

Offset: 0

R. H. Hardin, Apr 23 2013

nonn

Diagonal of A225010.

Number of unimodal maps [1..n]->[1..n+1], see example. - Joerg Arndt, May 10 2013

			Some solutions for n=3
..0..1..1....0..1..0....0..0..1....0..0..0....0..0..0....0..0..0....0..0..0
..1..1..1....0..1..0....1..1..1....0..0..0....0..0..0....0..1..0....0..0..1
..1..1..1....0..1..1....1..1..1....0..0..1....0..1..0....1..1..1....0..1..1
From _Joerg Arndt_, May 10 2013: (Start)
The a(2) = 9 unimodal maps [1,2]->[1,2,3] are
01:  [ 1 1 ]
02:  [ 1 2 ]
03:  [ 1 3 ]
04:  [ 2 1 ]
05:  [ 2 2 ]
06:  [ 2 3 ]
07:  [ 3 1 ]
08:  [ 3 2 ]
09:  [ 3 3 ]
(End)

G. C. Greubel and R. H. Hardin, Table of n, a(n) for n = 0..1000 (terms 1..51 from R. H. Hardin)

Cf. A088536 (unimodal maps [1..n]->[1..n]).

Cf. A183160, A371798.

a[n_] := Sum[Binomial[2d+n-1, n-1], {d, 0, n}]; Array[a, 30] (* Jean-François Alcover, Feb 17 2016, after Max Alekseyev *)

{ a(n) = polcoeff( (1+x+O(x^(2*n+1)))^(-n-1)/(1-x), 2*n) }

From Vaclav Kotesovec, May 22 2013: (Start)
Empirical: 4*n*(2*n-1)*(5*n-7)*a(n) = 2*(145*n^3 - 343*n^2 + 235*n - 48)*a(n-1) - 3*(3*n-4)*(3*n-2)*(5*n-2)*a(n-2).
a(n) ~ 3^(3*n+3/2)/(5*2^(2*n+1)*sqrt(Pi*n)). (End)
a(n) = A261668(n)+1.
a(n) = Sum_{d=0..n} binomial(2d+n-1,n-1). Also, a(n) is the coefficient of x^(2n) in (1+x)^(-n-1)/(1-x). - Max Alekseyev, Sep 14 2015
It appears that a(n) = Sum_{k = 0..2*n} (-1)^k*binomial(n+k,k). - Peter Bala, Oct 08 2021
From Seiichi Manyama, Apr 06 2024: (Start)
a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(3*n-2*k-1,n-2*k).
a(n) = [x^n] 1/((1+x^2) * (1-x)^(2*n)). (End)

a(0)=1 prepended by Alois P. Heinz, Feb 04 2017

A226031 Number A(n,k) of unimodal functions f:[n]->[k*n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 16, 22, 0, 1, 4, 36, 161, 130, 0, 1, 5, 64, 525, 1716, 791, 0, 1, 6, 100, 1222, 8086, 18832, 4900, 0, 1, 7, 144, 2360, 24616, 128248, 210574, 30738, 0, 1, 8, 196, 4047, 58730, 510664, 2072862, 2385644, 194634, 0

Offset: 0

Alois P. Heinz, May 23 2013

			Square array A(n,k) begins:
  1,   1,     1,      1,      1,       1, ...
  0,   1,     2,      3,      4,       5, ...
  0,   4,    16,     36,     64,     100, ...
  0,  22,   161,    525,   1222,    2360, ...
  0, 130,  1716,   8086,  24616,   58730, ...
  0, 791, 18832, 128248, 510664, 1505205, ...

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

Columns k=0-2 give: A000007, A088536, A226012.
Rows n=0-2 give: A000012, A001477, A016742.
Main diagonal gives: A227402.
Cf. A071920.

A:= (n, k)-> `if`(n=0, 1, add(binomial(n+2*j-1, 2*j), j=0..k*n-1)):
seq(seq(A(n, d-n), n=0..d), d=0..10);

A[n_, k_] := If[n==0, 1, Sum[Binomial[n + 2j - 1, 2j], {j, 0, k n - 1}]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

A(n,k) = Sum_{j=0..k*n-1} C(n+2*j-1,2*j), A(0,k) = 1.

A(n,k) = A071921(n,k*n).

cp's OEIS Frontend

A223718 Number of unimodal functions [1..n]->[0..2].

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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.

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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.

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A225006 Number of n X n 0..1 arrays with rows unimodal and columns nondecreasing.

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A226031 Number A(n,k) of unimodal functions f:[n]->[k*n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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