A131423 -id:A131423 - cp's OEIS frontend

A143371 Duplicate of A131423.

1, 8, 25, 56, 105, 176, 273, 400, 561, 760, 1001, 1288, 1625, 2016, 2465, 2976, 3553, 4200, 4921, 5720, 6601, 7568, 8625, 9776, 11025, 12376, 13833, 15400, 17081, 18880, 20801, 22848, 25025, 27336, 29785, 32376, 35113, 38000, 41041, 44240, 47601

Offset: 1

dead

A200838 T(n,k)=Number of 0..k arrays x(0..n+1) of n+2 elements without any two consecutive increases or two consecutive decreases.

Original entry on oeis.org

8, 25, 16, 56, 69, 32, 105, 194, 191, 64, 176, 435, 676, 529, 128, 273, 846, 1817, 2356, 1465, 256, 400, 1491, 4108, 7587, 8210, 4057, 512, 561, 2444, 8239, 19930, 31677, 28610, 11235, 1024, 760, 3789, 15128, 45465, 96690, 132263, 99700, 31113, 2048

Offset: 1

R. H. Hardin Nov 23 2011

Table starts
....8.....25......56......105.......176........273........400.........561
...16.....69.....194......435.......846.......1491.......2444........3789
...32....191.....676.....1817......4108.......8239......15128.......25953
...64....529....2356.....7587.....19930......45465......93472......177381
..128...1465....8210....31677.....96690.....250913.....577660.....1212729
..256...4057...28610...132263....469116....1384813....3570086.....8291391
..512..11235...99700...552247...2276028....7642875...22063924....56687801
.1024..31113..347434..2305835..11042700...42181611..136360286...387572529
.2048..86161.1210736..9627715..53576350..232803603..842739040..2649819955
.4096.238605.4219166.40199277.259938722.1284861277.5208328180.18116728573

			Some solutions for n=4 k=3
..1....2....3....0....1....1....2....1....3....3....3....1....2....0....1....1
..0....0....0....2....1....0....3....3....1....3....0....3....2....3....1....0
..0....0....2....2....0....3....0....0....1....2....1....3....2....0....1....1
..3....0....1....3....3....3....3....2....1....2....0....1....2....0....0....1
..3....3....3....0....3....0....1....2....1....1....3....3....3....2....2....3
..1....3....2....0....1....3....3....2....2....1....0....1....2....1....1....0

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A000079(n+2)
Column 2 is A098182(n+3)
Row 1 is A131423(n+1)

Empirical for columns:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 3*a(n-1) -a(n-2) +a(n-3)
k=3: a(n) = 4*a(n-1) -2*a(n-2) +a(n-3) -a(n-4)
k=4: a(n) = 5*a(n-1) -4*a(n-2) +3*a(n-3) -3*a(n-4) +a(n-5) -a(n-6)
k=5: a(n) = 6*a(n-1) -6*a(n-2) +3*a(n-3) -5*a(n-4) +3*a(n-5) -2*a(n-6) +a(n-7)
k=6: a(n) = 7*a(n-1) -9*a(n-2) +6*a(n-3) -9*a(n-4) +7*a(n-5) -7*a(n-6) +5*a(n-7) -2*a(n-8) +a(n-9)
k=7: a(n) = 8*a(n-1) -12*a(n-2) +6*a(n-3) -10*a(n-4) +12*a(n-5) -11*a(n-6) +11*a(n-7) -6*a(n-8) +3*a(n-9) -a(n-10)
Empirical for rows:
n=1: a(k) = (2/3)*k^3 + 3*k^2 + (10/3)*k + 1
n=2: a(k) = (5/12)*k^4 + (19/6)*k^3 + (79/12)*k^2 + (29/6)*k + 1
n=3: a(k) = (4/15)*k^5 + (17/6)*k^4 + (28/3)*k^3 + (73/6)*k^2 + (32/5)*k + 1
n=4: a(k) = (61/360)*k^6 + (93/40)*k^5 + (779/72)*k^4 + (521/24)*k^3 + (1801/90)*k^2 + (239/30)*k + 1
n=5: a(k) = (34/315)*k^7 + (163/90)*k^6 + (1981/180)*k^5 + (557/18)*k^4 + (7807/180)*k^3 + (1361/45)*k^2 + (333/35)*k + 1
n=6: a(k) = (277/4032)*k^8 + (1375/1008)*k^7 + (4933/480)*k^6 + (2723/72)*k^5 + (14161/192)*k^4 + (11197/144)*k^3 + (216211/5040)*k^2 + (929/84)*k + 1
n=7: a(k) = (124/2835)*k^9 + (1123/1120)*k^8 + (244/27)*k^7 + (1991/48)*k^6 + (57133/540)*k^5 + (74183/480)*k^4 + (291427/2268)*k^3 + (9739/168)*k^2 + (568/45)*k + 1

A182222 Number T(n,k) of standard Young tableaux of n cells and height >= k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 10, 10, 9, 4, 1, 26, 26, 25, 16, 5, 1, 76, 76, 75, 56, 25, 6, 1, 232, 232, 231, 197, 105, 36, 7, 1, 764, 764, 763, 694, 441, 176, 49, 8, 1, 2620, 2620, 2619, 2494, 1785, 856, 273, 64, 9, 1, 9496, 9496, 9495, 9244, 7308, 3952, 1506, 400, 81, 10, 1

Offset: 0

Alois P. Heinz, Apr 19 2012

Also number of self-inverse permutations in S_n with longest increasing subsequence of length >= k. T(4,3) = 4: 1234, 1243, 1324, 2134; T(3,0) = T(3,1) = 4: 123, 132, 213, 321; T(5,3) = 16: 12345, 12354, 12435, 12543, 13245, 13254, 14325, 14523, 15342, 21345, 21354, 21435, 32145, 34125, 42315, 52341.

			T(4,3) = 4, there are 4 standard Young tableaux of 4 cells and height >= 3:
  +---+   +------+   +------+   +------+
  | 1 |   | 1  2 |   | 1  3 |   | 1  4 |
  | 2 |   | 3 .--+   | 2 .--+   | 2 .--+
  | 3 |   | 4 |      | 4 |      | 3 |
  | 4 |   +---+      +---+      +---+
  +---+
Triangle T(n,k) begins:
    1;
    1,   1;
    2,   2,   1;
    4,   4,   3,   1;
   10,  10,   9,   4,   1;
   26,  26,  25,  16,   5,   1;
   76,  76,  75,  56,  25,   6,  1;
  232, 232, 231, 197, 105,  36,  7,  1;
  764, 764, 763, 694, 441, 176, 49,  8,  1;
  ...

Alois P. Heinz, Rows n = 0..50, flattened
Wikipedia, Involution (mathematics)
Wikipedia, Young tableau

Columns 0-10 give: A000085, A000085 (for n>0), A001189, A218263, A218264, A218265, A218266, A218267, A218268, A218269, A218262.
Diagonal and lower diagonals give: A000012, A000027(n+1), A000290(n+1) for n>0, A131423(n+1) for n>1.
T(2n,n) gives A318289.
Cf. A047884, A049400, A182172.

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j+
       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l) option remember;
      `if`(n=0, h(l), `if`(i<1, 0, `if`(i=1, h([l[], 1$n]),
        g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
    end:
T:= (n, k)-> g(n, n, []) -`if`(k=0, 0, g(n, k-1, [])):
seq(seq(T(n, k), k=0..n), n=0..12);

h[l_] := Module[{n = Length[l]}, Sum[i, {i, l}]! / Product[ Product[1 + l[[i]] - j + Sum [If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := g[n, i, l] = If[n == 0, h[l], If[i < 1, 0, If[i == 1, h[Join[l, Array[1&, n]]], g [n, i-1, l] + If[i > n, 0, g[n-i, i, Append[l, i]]]]]];
t[n_, k_] := g[n, n, {}] - If[k == 0, 0, g[n, k-1, {}]];
Table[Table[t[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)

T(n,k) = A182172(n,n) - A182172(n,k-1) for k>0, T(n,0) = A182172(n,n).

A143368 Triangle read by rows: T(n,k) is the Wiener index of a k X n grid (i.e., P_k X P_n, where P_m is the path graph on m vertices; 1 <= k <= n).

Original entry on oeis.org

0, 1, 8, 4, 25, 72, 10, 56, 154, 320, 20, 105, 280, 570, 1000, 35, 176, 459, 920, 1595, 2520, 56, 273, 700, 1386, 2380, 3731, 5488, 84, 400, 1012, 1984, 3380, 5264, 7700, 10752, 120, 561, 1404, 2730, 4620, 7155, 10416, 14484, 19440

Offset: 1

Emeric Deutsch, Sep 05 2008

The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.

This is the lower triangular half of a symmetric square array.

			Presentation as symmetric square array starts:
======================================================
n\k|   1   2    3    4    5    6     7     8     9
---|--------------------------------------------------
1  |   0   1    4   10   20   35    56    84   120 ...
2  |   1   8   25   56  105  176   273   400   561 ...
3  |   4  25   72  154  280  459   700  1012  1404 ...
4  |  10  56  154  320  570  920  1386  1984  2730 ...
5  |  20 105  280  570 1000 1595  2380  3380  4620 ...
6  |  35 176  459  920 1595 2520  3731  5264  7155 ...
7  |  56 273  700 1386 2380 3731  5488  7700 10416 ...
8  |  84 400 1012 1984 3380 5264  7700 10752 14484 ...
9  | 120 561 1404 2730 4620 7155 10416 14484 19440 ...
... - _Andrew Howroyd_, May 27 2017
T(2,2)=8 because in a square we have four distances equal to 1 and two distances equal to 2.
T(2,1)=1 because on the path graph on two vertices there is one distance equal to 1.
T(3,2)=25 because on the P(2) X P(3) graph there are 7 distances equal to 1, 6 distances equal to 2 and 2 distances equal to 3, with 7*1 + 6*2 + 2*3 = 25.
Triangle starts: 0; 1,8; 4,25,72; 10,56,154,320;

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150).
A. Graovac and T. Pisanski, On the Wiener index of a graph, J. Math. Chem., 8 (1991), 53-62.
B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Wiener Index

Cf. A180569 (row 3), A131423 (row 2).
Main diagonal is A143945.
Cf. A245826.

T:=proc(n, k) options operator, arrow: (1/6)*k*n*(n+k)*(k*n-1) end proc: for n to 9 do seq(T(n,k),k=1..n) end do; # yields sequence in triangular form

Table[k n (n + k) (k n - 1)/6, {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, May 28 2017 *)

T(n,k)=k*n*(n+k)*(k*n-1)/6;
for (n=1,8,for(k=1,8,print1(T(n,k),", "));print) \\ Andrew Howroyd, May 27 2017

T(n,k) = k*n*(n+k)*(k*n-1)/6 (k, n >= 1).

A223544 T(n, k) = n*k - 1.

Original entry on oeis.org

0, 1, 3, 2, 5, 8, 3, 7, 11, 15, 4, 9, 14, 19, 24, 5, 11, 17, 23, 29, 35, 6, 13, 20, 27, 34, 41, 48, 7, 15, 23, 31, 39, 47, 55, 63, 8, 17, 26, 35, 44, 53, 62, 71, 80, 9, 19, 29, 39, 49, 59, 69, 79, 89, 99, 10, 21, 32, 43, 54, 65, 76, 87, 98, 109, 120, 11, 23, 35, 47, 59, 71, 83, 95, 107, 119, 131, 143

Offset: 1

Richard R. Forberg, Jul 19 2013

Previous name was: Triangle T(n,k), 0 < k <= n, T(n,1) = n - 1, T(n,k) = T(n,k-1) + n; read by rows.
This simple triangle arose analyzing f(x) = x/(n + e^(c/x)), for n <> 0. f(x) converges towards a rational number for large values of x, if x is rational. T(n+1,k)/(n+1)^2 equals the fractional portion of f(x) if x is large and restricted to the positive integers, c = 1 and n>=1, whereby the value of the fractional portion changes on a cycle with period n+1 (as k goes from 1 to n+1) for each n in the denominator of f(x). Other, somewhat similar triangles (or repeating fractional patterns) arise with other rational values of n or c, or other rational increments of x (even if a large irrational initial value of x is used).
Let S(n) = row sums = Sum(k>=1, T(n,k)), then:
S(n) = A077414(n);  S(n)/(n+2) = A000217(n); S(n)/n = A000096(n);
Let Sq(n) = sum of squares of row elements = Sum(k>=1, T(n,k)^2), then:
   Sq(n)/n^2 - 1/n = A058373(n)
Let D(n) = diagonal sums = Sum(k>=1, T(n-k+1, k)) then:
   D(2n) = A131423(n); D(2n-1) = 2/3*n^3 + 1/2*n^2 - 7/6*n;
   D(2n) - D(2n-1) = A000217(n); D(2n+1) - D(2n) = A115067(n);
   D(2n+2) - D(2n)= A056220(n+1);  D(2n+1) - D(2n -1) = A014106(n).
Equals A144204 with the first column of negative ones removed. - Georg Fischer, Jul 26 2023

			Triangle begins as:
0;
1,  3;
2,  5,  8;
3,  7, 11, 15;
4,  9  14, 19, 24;
5, 11, 17, 23, 29, 35;
6, 13, 20, 27, 34, 41, 48;
7, 15, 23, 31, 39, 47, 55, 63;
8, 17, 26, 35, 44, 53, 62, 71, 80;

Boris Putievskiy, Rows n = 1..140 of triangle, flattened

Cf. A075362, A144204.

Also note: T(n+1,k) = T(n,k)+ k, and T(n,n) = n^2 - 1.
a(n) = A075362(n)-1; a(n)=i(t+1)-1, where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Jul 24 2013
T(n, k) = n*k - 1. - Georg Fischer, Jul 26 2023

Simpler name from Georg Fischer, Jul 26 2023

A369324 Array read by ascending antidiagonals: A(n,k) is the number of words of length n on an alphabet [k], avoiding 120 and 210, and sortable by a stack of depth 2, where k >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 8, 9, 4, 1, 0, 1, 16, 25, 16, 5, 1, 0, 1, 32, 65, 56, 25, 6, 1, 0, 1, 64, 161, 176, 105, 36, 7, 1, 0, 1, 128, 385, 512, 385, 176, 49, 8, 1, 0, 1, 256, 897, 1408, 1281, 736, 273, 64, 9, 1, 0, 1, 512, 2049, 3712, 3969, 2752, 1281, 400, 81, 10, 1

Offset: 0

Stefano Spezia, Jan 20 2024

			The array begins:
  0, 1,  1,   1,   1,    1, ...
  0, 1,  2,   3,   4,    5, ...
  0, 1,  4,   9,  16,   25, ...
  0, 1,  8,  25,  56,  105, ...
  0, 1, 16,  65, 176,  385, ...
  0, 1, 32, 161, 512, 1281, ...
  ...

Toufik Mansour, Howard Skogman, and Rebecca Smith, Sorting inversion sequences, arXiv:2401.06662 [math.CO], 2024. See Theorem 3.18 at page 10.

Cf. A000004 (k=0), A000012 (k=1), A000079 (k=2), A002064 (k=3), A340257 (k=4).
Cf. A000290 (n=2), A001477 (n=1), A057427 (n=0), A131423 (n=3), A164039.
Cf. A000035, A369325 (main diagonal), A369326.

A[n_,k_]:=(1-(-1)^k)/2+2^n Sum[Binomial[n+k-3-2i,n-1],{i,0,Floor[(k-2)/2]}]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

A(n,k) = A000035(k) + 2^n*Sum_{i=0..floor((k-2)/2)} binomial(n + k - 3 - 2*i, n - 1).

Sum_{k=0..n} A(n-k,k) = A164039(n-1).

A131422 (A000012 * A127773) + (A127773 * A000012) - A000012.

Original entry on oeis.org

1, 3, 5, 6, 8, 11, 10, 12, 15, 19, 15, 17, 20, 24, 29, 21, 23, 26, 30, 35, 41, 28, 30, 33, 37, 42, 48, 55, 36, 38, 41, 45, 50, 56, 63, 71, 45, 47, 50, 54, 59, 65, 72, 80, 89, 55, 57, 60, 64, 69, 75, 82, 90, 99, 109

Offset: 1

Gary W. Adamson, Jul 10 2007

Left border = the triangular series, A000217. Right border = A028387, (1, 5, 11, 19, 29, 41, 55, 71, ...). Row sums = A131423: (1, 8, 25, 56, 105, 176, 273, ...).

			First few rows of the triangle are:
   1;
   3,  5;
   6,  8, 11;
  10, 12, 15, 19;
  15, 17, 20, 24, 29;
  21, 23, 26, 30, 35, 41;
  28, 30, 33, 37, 42, 48, 55;
  ...

Cf. A127773, A000217, A028387, A131423.

T:=proc(n,k) options operator, arrow: (1/2)*n*(n+1)+(1/2)*k*(k+1)-1 end proc: for n to 10 do seq(T(n,k),k=1..n) end do; # yields sequence in triangular form - Emeric Deutsch, Sep 06 2008

(A000012 * A127773) + (A127773 * A000012) - A000012 as infinite lower triangular matrices.

T(n,k) = (n^2 + n + k^2 + k - 2)/2 (1 <= k <= n). - Emeric Deutsch, Sep 06 2008

A143370 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the grid P_2 x P_n (1 <= k <= n). P_m is the path graph on m vertices.

Original entry on oeis.org

1, 4, 2, 7, 6, 2, 10, 10, 6, 2, 13, 14, 10, 6, 2, 16, 18, 14, 10, 6, 2, 19, 22, 18, 14, 10, 6, 2, 22, 26, 22, 18, 14, 10, 6, 2, 25, 30, 26, 22, 18, 14, 10, 6, 2, 28, 34, 30, 26, 22, 18, 14, 10, 6, 2, 31, 38, 34, 30, 26, 22, 18, 14, 10, 6, 2, 34, 42, 38, 34, 30, 26, 22, 18, 14, 10, 6, 2

Offset: 1

Emeric Deutsch, Sep 05 2008

Sum of entries in row n = n(2n-1) = A000384(n).
The entries in row n are the coefficients of the Wiener polynomial of the grid P_2 x P_n.
Sum_{k=1..n} k*T(n,k) = A131423(n) = the Wiener index of the grid P_2 x P_n.
The average of all distances in the grid P_2 x P_n is (n+2)/3.

			T(2,1)=4 because in the graph P_2 x P_2 (a square) we have 4 distances equal to 1.
Triangle starts:
   1;
   4,  2;
   7,  6,  2;
  10, 10,  6,  2;
  13, 14, 10,  6,  2;

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Cf. A000384.

G:=q*z*(1+2*z+q*z)/((1-z)^2*(1-q*z)): Gser:= simplify(series(G,z=0,15)): for n to 12 do p[n]:=sort(coeff(Gser,z,n)) end do: for n to 12 do seq(coeff(p[n],q, j),j=1..n) end do; # yields sequence in triangular form

G.f. = G(q,z) = qz(1+2z+qz)/((1-qz)(1-z)^2).

A180865 Square array read by antidiagonals: T(m,n) is the Wiener index of the stacked book graph B(m,n) (m>=1, n>=1). B(m,n) is defined as the graph Cartesian product S(m+1) x P(n), where S(m+1) is the star graph on m+1 nodes and P(n) is the path graph on n nodes. The Wiener index of a connected graph is the sum of distances between all unordered pairs of nodes in the graph.

Original entry on oeis.org

1, 4, 8, 9, 25, 25, 16, 52, 72, 56, 25, 89, 145, 154, 105, 36, 136, 244, 304, 280, 176, 49, 193, 369, 506, 545, 459, 273, 64, 260, 520, 760, 900, 884, 700, 400, 81, 337, 697, 1066, 1345, 1451, 1337, 1012, 561, 100, 424, 900, 1424, 1880, 2160, 2184, 1920, 1404, 760

Offset: 1

Emeric Deutsch, Sep 28 2010

T(1,n) = A131423(n).

T(2,n) = A180569(n).

			T(2,1)=4 because B(2,1) reduces to the path graph P(3) which has 2 pairs of nodes at distance 1 and 1 pair at distance 2.
Square array T(m,n) begins:
1, 8, 25, 56, 105, ...
4, 25, 72, 154, 280, ...
9, 52, 145, 304, 545, ...
16, 89, 244, 506, 900, ...

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
Eric Weisstein's World of Mathematics, Stacked Book Graph.

Cf. A131423, A180569.

T := proc (m, n) options operator, arrow: (1/6)*n*(n^2-(m+1)^2+m*n*(m*n+2*n+6*m)) end proc: for n to 10 do seq(T(n+1-j, j), j = 1 .. n) end do; # yields sequence in triangular form

T(m,n) = (1/6)n[n^2-(m+1)^2+mn(mn+6m+2n)].

The Wiener polynomial p[n](t) of the graph B(m,n) satisfies the recurrence relation p[n] = p[n-1]+mt+(1/2)m(m-1)t^2+[t+mt+2mt^2+m(m-1)t^3]*sum(t^j,j=0..n-2).

A232535 Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2n,2k) + binomial(2n+1,2k))/2.

Original entry on oeis.org

1, 1, 2, 1, 8, 3, 1, 18, 25, 4, 1, 32, 98, 56, 5, 1, 50, 270, 336, 105, 6, 1, 72, 605, 1320, 891, 176, 7, 1, 98, 1183, 4004, 4719, 2002, 273, 8, 1, 128, 2100, 10192, 18590, 13728, 4004, 400, 9, 1, 162, 3468, 22848, 59670, 68068, 34476, 7344, 561, 10, 1, 200, 5415

Offset: 0

Philippe Deléham, Nov 25 2013

Sum_{k=0..n}T(n,k)*x^k = A164111(n), A000012(n), A002001(n), A001653(n+1), A001019(n), A166965(n) for x =-1, 0, 1, 2, 4, 9 respectively.

			Triangle begins:
1
1, 2
1, 8, 3
1, 18, 25, 4
1, 32, 98, 56, 5
1, 50, 270, 336, 105, 6
1, 72, 605, 1320, 891, 176, 7
1, 98, 1183, 4004, 4719, 2002, 273, 8
1, 128, 2100, 10192, 18590, 13728, 4004, 400, 9

Cf. A086645, A091042
Cf. Columns : A000012, A001105, A180324 ; Diagonals: A000027, A131423
Cf. T(2*n,n): A228329, Row sums : A002001

T := (n,k) -> binomial(2*n, 2*k)*(2*n+1-k)/(2*n+1-2*k);
seq(seq(T(n,k), k=0..n), n=0..9); # Peter Luschny, Nov 26 2013

Flatten[Table[(Binomial[2n,2k]+Binomial[2n+1,2k])/2,{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jul 05 2015 *)

G.f.: (1-x)/(1-2*x*(1+y)+x^2*(1-y)^2).
T(n,k) = 2*T(n-1,k)+2*T(n-1,k-1)+2*T(n-2,k-1)-T(n-2,k)-T(n-2,k-2), T(0,0)=T(1,0)=1, T(1,1)=2, T(n,k)=0 if k<0 or if k>n.
T(n,k) = (A086645(n,k) + A091042(n,k))/2.
T(n,k) = binomial(2*n,2*k)*(2*n+1-k)/(2*n+1-2*k). - Peter Luschny, Nov 26 2013

cp's OEIS Frontend

A143371 Duplicate of A131423.

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A200838 T(n,k)=Number of 0..k arrays x(0..n+1) of n+2 elements without any two consecutive increases or two consecutive decreases.

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A182222 Number T(n,k) of standard Young tableaux of n cells and height >= k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A143368 Triangle read by rows: T(n,k) is the Wiener index of a k X n grid (i.e., P_k X P_n, where P_m is the path graph on m vertices; 1 <= k <= n).

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A223544 T(n, k) = n*k - 1.

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A369324 Array read by ascending antidiagonals: A(n,k) is the number of words of length n on an alphabet [k], avoiding 120 and 210, and sortable by a stack of depth 2, where k >= 0.

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A131422 (A000012 * A127773) + (A127773 * A000012) - A000012.

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A143370 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the grid P_2 x P_n (1 <= k <= n). P_m is the path graph on m vertices.

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A232535 Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2n,2k) + binomial(2n+1,2k))/2.