A357368 -id:A357368 - cp's OEIS frontend

A075435 T(n,k) = right- or upward-moving paths connecting opposite corners of an n X n chessboard, visiting the diagonal at k points between start and finish.

2, 6, 4, 20, 24, 8, 70, 116, 72, 16, 252, 520, 456, 192, 32, 924, 2248, 2496, 1504, 480, 64, 3432, 9520, 12624, 9728, 4480, 1152, 128, 12870, 39796, 60792, 56400, 33440, 12480, 2688, 256, 48620, 164904, 283208, 304704, 218720, 105600, 33152, 6144

Offset: 2

Wouter Meeussen, Sep 15 2002

If it is required that the paths stay at the same side of the diagonal between intermediate points, then the count of intermediate points becomes an exact count of crossings and one gets table A039598.

T is the convolution triangle of the central binomial coefficients. - Peter Luschny, Oct 19 2022

			{2},
{6, 4},
{20, 24, 8},
{70, 116, 72, 16},
{252, 520, 456, 192, 32},
...

Row sums give A075436.

Cf. A039598, A000108.

# Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
PMatrix(10, n -> binomial(2*n, n)); # Peter Luschny, Oct 19 2022

Table[Table[Plus@@Apply[Times, Compositions[n-1-k, k]+1 /. i_Integer->Binomial[2i, i], {1}], {k, 1, n-1}], {n, 2, 12}]

T(n,m):=sum(k/n*binomial(2*n-k-1,n-1)*2^k*binomial(k-1,m-1),k,m,n); /* Vladimir Kruchinin, Mar 30 2011 */

@cached_function
def T(k,n):
    if k==n: return 2^n
    if k==0: return 0
    return sum(binomial(2*i,i)*T(k-1,n-i) for i in (1..n-k+1))
A075435 = lambda n,k: T(k,n)
for n in (1..9): print([A075435(n,k) for k in (1..n)]) # Peter Luschny, Mar 12 2016

G.f.: [2*x*c(x)/(1-x*c(x))]^m=sum(n>=m T(n,m)*x^n) where c(x) is the g.f. of A000108, also T(n,m)=sum(k=m..n, k/n*binomial(2*n-k-1,n-1)*2^k*binomial(k-1,m-1)), n>=m>0. [Vladimir Kruchinin, Mar 30 2011]

A105495 Triangle read by rows: T(n,k) is the number of compositions of n into k parts when parts equal to q are of q^2 kinds.

Original entry on oeis.org

1, 4, 1, 9, 8, 1, 16, 34, 12, 1, 25, 104, 75, 16, 1, 36, 259, 328, 132, 20, 1, 49, 560, 1134, 752, 205, 24, 1, 64, 1092, 3312, 3338, 1440, 294, 28, 1, 81, 1968, 8514, 12336, 7815, 2456, 399, 32, 1, 100, 3333, 19800, 39572, 35004, 15765, 3864, 520, 36, 1, 121, 5368

Offset: 1

Emeric Deutsch, Apr 10 2005

Triangle T(n,k)=
1. Riordan Array (1,(x+x^2)/(1-x)^3) without first column.
2. Riordan Array ((1+x)/(1-x)^3,(x+x^2)/(1-x)^3) numbering triangle (0,0).
[Vladimir Kruchinin, Nov 25 2011]
Triangle T(n,k), 1<=k<=n, given by (0, 4, -7/4, 17/28, -32/119, 7/17, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 20 2012
T is the convolution triangle of the squares (A000290). - Peter Luschny, Oct 19 2022

			T(3,2)=8 because we have (1,2),(1,2'),(1,2"),(1,2'"),(2,1),(2',1),(2",1) and (2'",1).
Triangle begins:
  1;
  4,1;
  9,8,1;
  16,34,12,1;
  25,104,75,16,1;
  ...
Triangle (0, 4, -7/4, 17/28, -32/119, 7/17, 0, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins :
  1
  0, 1
  0, 4, 1
  0, 9, 8, 1
  0, 16, 34, 12, 1
  0, 25, 104, 75, 16, 1
  ...

Cf. A000290, A084938.

Row sums yield A033453.

G:=t*z*(1+z)/((1-z)^3-t*z*(1+z)): Gser:=simplify(series(G,z=0,13)): for n from 1 to 12 do P[n]:=coeff(Gser,z^n) od: for n from 1 to 11 do seq(coeff(P[n],t^k), k=1..n) od; # yields sequence in triangular form
# Alternatively:
T := proc(k,n) option remember;
if k=n then 1 elif k=0 then 0 else add(i^2*T(k-1,n-i), i=1..n-k+1) fi end:
A105495 := (n,k) -> T(k,n):
for n from 1 to 9 do seq(A105495(n,k), k=1..n) od; # Peter Luschny, Mar 12 2016
# Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
PMatrix(10, n -> n^2); # Peter Luschny, Oct 19 2022

nn=8;a=(x+x^2)/(1-x)^3;CoefficientList[Series[1/(1-y a),{x,0,nn}],{x,y}]//Grid  (* Geoffrey Critzer, Aug 31 2012 *)

T(n,k):=sum(binomial(k,i)*binomial(n+2*k-i-1,3*k-1),i,0,n-k); /* Vladimir Kruchinin, Nov 25 2011 */

@cached_function
def T(k,n):
    if k==n: return 1
    if k==0: return 0
    return sum(i^2*T(k-1,n-i) for i in (1..n-k+1))
A105495 = lambda n,k: T(k, n)
for n in (0..6): print([A105495(n, k) for k in (0..n)]) # Peter Luschny, Mar 12 2016

G.f.: t*z*(1+z)/((1-z)^3-t*z*(1+z)).
From Vladimir Kruchinin, Nov 25 2011: (Start)
G.f.: ((x+x^2)/(1-x)^3)^k = Sum_{n>=k} T(n,k)*x^n.
T(n,k) = Sum{i=0..n-k} binomial(k,i)*binomial(n+2*k-i-1,3*k-1). (End)

A127898 Inverse of Riordan array (1/(1+x)^3, x/(1+x)^3).

Original entry on oeis.org

1, 3, 1, 12, 6, 1, 55, 33, 9, 1, 273, 182, 63, 12, 1, 1428, 1020, 408, 102, 15, 1, 7752, 5814, 2565, 760, 150, 18, 1, 43263, 33649, 15939, 5313, 1265, 207, 21, 1, 246675, 197340, 98670, 35880, 9750, 1950, 273, 24, 1

Offset: 0

Paul Barry, Feb 04 2007

The convolution triangle of A001764 (number of ternary trees). - Peter Luschny, Oct 09 2022

			Triangle begins:
1,
3, 1,
12, 6, 1,
55, 33, 9, 1,
273, 182, 63, 12, 1,
1428, 1020, 408, 102, 15, 1,
7752, 5814, 2565, 760, 150, 18, 1,
43263, 33649, 15939, 5313, 1265, 207, 21, 1,
246675, 197340, 98670, 35880, 9750, 1950, 273, 24, 1,
1430715, 1170585, 610740, 237510, 71253, 16443, 2842, 348, 27, 1,
8414640, 7012200, 3786588, 1553472, 503440, 129456, 26040, 3968, 432, 30, 1

G. C. Greubel, Rows n=0..100 of triangle, flattened
Paul Drube, Generalized Path Pairs and Fuss-Catalan Triangles, arXiv:2007.01892 [math.CO], 2020. See Figure 4 p. 8.

First column is A001764(n+1).
Row sums are A047099.
Inverse of A127895.

Flat(List([0..10],n->List([0..n],k->(k+1)/(n+1)*Binomial(3*n+3,n-k)))); # Muniru A Asiru, Apr 30 2018

/* As triangle: */ [[(k+1)/(n+1)*Binomial(3*n+3,n-k): k in [0..n]]: n in [0..8]];  // Bruno Berselli, Jan 17 2013

# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> binomial(3*n, n)/(2*n+1)); # Peter Luschny, Oct 09 2022

Table[If[k == 0, Binomial[3*n, n-k]/(2*n+1), ((k+1)/n)*Binomial[3*n, n-k -1]], {n,1,10}, {k,0,n-1}]//Flatten (* G. C. Greubel, Apr 29 2018 *)

for(n=1,10, for(k=0,n-1, print1(if(k==0, binomial(3*n, n-k)/( 2*n +1), ((k+1)/n)*binomial(3*n, n-k-1)), ", "))) \\ G. C. Greubel, Apr 29 2018

T(n,k) = (k+1)/(n+1)*binomial(3*n+3,n-k). - Vladimir Kruchinin, Jan 17 2013
G.f.: 1/(-y + 1/(-1 + (2*sin(1/3 *arcsin((3*sqrt(3*x))/2)))/(
  sqrt(3*x))))/x.  - Vladimir Kruchinin, Feb 14 2023

A206306 Riordan array (1, x/(1-3x+2x^2)).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 7, 6, 1, 0, 15, 23, 9, 1, 0, 31, 72, 48, 12, 1, 0, 63, 201, 198, 82, 15, 1, 0, 127, 522, 699, 420, 125, 18, 1, 0, 255, 1291, 2223, 1795, 765, 177, 21, 1, 0, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1

Offset: 0

Philippe Deléham, Feb 06 2012

The convolution triangle of the Mersenne numbers A000225. - Peter Luschny, Oct 09 2022

			Triangle begins:
  1;
  0,    1;
  0,    3,    1;
  0,    7,    6,     1;
  0,   15,   23,     9,     1;
  0,   31,   72,    48,    12,     1;
  0,   63,  201,   198,    82,    15,    1;
  0,  127,  522,   699,   420,   125,   18,    1;
  0,  255, 1291,  2223,  1795,   765,  177,   21,   1;
  0,  511, 3084,  6562,  6768,  3840, 1260,  238,  24,  1;
  0, 1023, 7181, 18324, 23276, 16758, 7266, 1932, 308, 27,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Columns: A000007, A000225 (Mersenne numbers), A045618, A055582.
Cf. A008585, A009545, A084938, A062725, A110441, A204089, A204091.
Cf. A045741, A078020, A244038.

function T(n,k) // T = A206306
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  elif k eq 0 then return 0;
  else return 3*T(n-1, k) +T(n-1, k-1) -2*T(n-2, k);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 20 2022

# Uses function PMatrix from A357368.
PMatrix(10, n -> 2^n - 1); # Peter Luschny, Oct 09 2022

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==n, 1, If[k==0, 0, 3*T[n- 1, k] +T[n-1, k-1] -2*T[n-2, k]]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 20 2022 *)

def T(n,k): # T = A206306
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    elif (k==0): return 0
    else: return 3*T(n-1, k) +T(n-1, k-1) -2*T(n-2, k)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 20 2022

Triangle T(n,k), read by rows, given by (0, 3, -2/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Diagonals sums are even-indexed Fibonacci numbers.
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A204089(n), A204091(n) for x = 0, 1, 2 respectively.
G.f.: (1-3*x+2*x^)/(1-(3+y)*x+2*x^2).
From Philippe Deléham, Nov 17 2013; corrected Feb 13 2020: (Start)
T(n, n) = 1.
T(n+1, n) = 3n = A008585(n).
T(n+2, n) = A062725(n).
T(n,k) = 3*T(n-1,k)+T(n-1,k-1)-2*T(n-2,k), T(0,0)=T(1,1)=T(2,2)=1, T(1,0)=T(2,0)=0, T(2,1)=3, T(n,k)=0 if k<0 or if k>n. (End)
From G. C. Greubel, Dec 20 2022: (Start)
Sum_{k=0..n} (-1)^k*T(n,k) = [n=1] - A009545(n).
Sum_{k=0..n} (-2)^k*T(n,k) = [n=1] + A078020(n+1).
T(2*n, n+1) = A045741(n+2), n >= 0.
T(2*n+1, n+1) = A244038(n). (End)

A357583 Triangle read by rows. Convolution triangle of the Bell numbers.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 5, 4, 1, 0, 15, 14, 6, 1, 0, 52, 50, 27, 8, 1, 0, 203, 189, 113, 44, 10, 1, 0, 877, 764, 471, 212, 65, 12, 1, 0, 4140, 3311, 2013, 974, 355, 90, 14, 1, 0, 21147, 15378, 8951, 4440, 1790, 550, 119, 16, 1, 0, 115975, 76418, 41745, 20526, 8727, 3027, 805, 152, 18, 1

Offset: 0

Peter Luschny, Oct 05 2022

			Triangle T(n, k) starts:
  [0] 1;
  [1] 0,     1;
  [2] 0,     2,     1;
  [3] 0,     5,     4,    1;
  [4] 0,    15,    14,    6,    1;
  [5] 0,    52,    50,   27,    8,    1;
  [6] 0,   203,   189,  113,   44,   10,   1;
  [7] 0,   877,   764,  471,  212,   65,  12,   1;
  [8] 0,  4140,  3311, 2013,  974,  355,  90,  14,  1;
  [9] 0, 21147, 15378, 8951, 4440, 1790, 550, 119, 16, 1;

Cf. A000110, A129247 (row sums), A007311, A357584 (central terms).

# Using function PMatrix from A357368.
PMatrix(10, combinat[bell]);

Conjecture: row polynomials are x*R(n,1) for n > 0 where R(n,k) = R(n-1,k+1) + x*R(n-1,1)*R(1,k) for n > 1, k > 0 with R(1,k) = Bell(k) for k > 0. The same recursion seems to work for self-convolution of any other sequence. - Mikhail Kurkov, Apr 05 2025

A128627 Triangle read by rows. Convolution triangle based on A002865.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 2, 3, 0, 1, 2, 5, 3, 4, 0, 1, 4, 6, 9, 4, 5, 0, 1, 4, 13, 12, 14, 5, 6, 0, 1, 7, 16, 28, 20, 20, 6, 7, 0, 1, 8, 30, 39, 50, 30, 27, 7, 8, 0, 1, 12, 40, 78, 76, 80, 42, 35, 8, 9, 0, 1, 14, 66, 115, 161, 130, 119, 56, 44, 9, 10, 0, 1

Offset: 1

Alford Arnold, Mar 22 2007

Triangular array illustrating the application of cyclic partitions to the computation of partitions of an integer into parts of k kinds (cf. A060850).
The array is constructed by summing sequences associated with each cyclic partition as indicated below: (n' here denotes the sum of preceding sequences).
    1     2     3
   1     3     6
  4'     2     5     9
    1     2     3     4
   1     4     9    16
  5'     2     6    12    20
    1     2     3     4     5     6     7     8     9
   1     4     9    16    25    36    49    64    81
   1     3     6    10    15    21    28    36    45
  1     4    10    20    35    56    84   120   165
  6'     4    13    28    50    80   119   168   228   300
    1     2     3     4     5     6     7     8     9
   1     4     9    16    25    36    49    64    81
   1     4     9    16    25    36    49    64    81
  1     6    18    40    75   126   196   288   405
  7'     4    16    39    76   130   204   301   424   576
    1     2     3     4     5     6     7     8     9
   1     4     9    16    25    36    49    64    81
   1     4     9    16    25    36    49    64    81
   1     3     6    10    15    21    28    36    45
  1     6    18    40    75   126   196   288   405
  1     6    18    40    75   126   196   288   405
 1     5    15    35    70   126   210   330   495
  8'     7    30    78   161   290   477   735  1078  1521

			The diagonal 9th diagonal of A060850 is 22 185 810 2580 6765 ... and can be computed from a(n) and A007318 as illustrated:
   1
   0    1
   1    0    1
   1    2    0    1
   2    2    3    0
   2    5    3    4
   4    6    9    4
   4   13   12   14
   7   16   28   20
       30   39   50
            78   76
                161
times
   1
   1    9
   1    8   45
   1    7   36  165
   1    6   28  120
   1    5   21   84
   1    4   15   56
   1    3   10   35
   1    2    6   20
        1    3   10
             1    4
                  1
yields
   1
   0    9
   1    0   45
   1   14    0  165
   2   12   84    0
   2   25   63  336
   4   24  135  224
   4   39  120  490
   7   32  168  400
       30  117  500
            78  304
                161
summing to
  22  185  810 2580 ...
Triangle T(n, k) starts:
  [ 1] 1;
  [ 2] 0,  1;
  [ 3] 1,  0,  1;
  [ 4] 1,  2,  0,  1;
  [ 5] 2,  2,  3,  0,  1;
  [ 6] 2,  5,  3,  4,  0,  1;
  [ 7] 4,  6,  9,  4,  5,  0,  1;
  [ 8] 4, 13, 12, 14,  5,  6,  0,  1;
  [ 9] 7, 16, 28, 20, 20,  6,  7,  0,  1;
  [10] 8, 30, 39, 50, 30, 27,  7,  8,  0,  1;

Cf. A002865, A060850, A007318.

# Using function A002865 and function PMatrix from A357368.
A128627Triangle := proc(dim) local M, Row, r;
M := PMatrix(dim, n -> A002865(n-1));
Row := r -> convert(linalg:-row(M, r), list)[2..r];
for r from 2 to dim do lprint(Row(r)) od end:
A128627Triangle(11); # Peter Luschny, Oct 03 2022

New name by Peter Luschny, Oct 03 2022

A188285 Riordan matrix ( (1-2x)/(1-2x-x^2), (x-2x^2)/(1-2x-x^2) ).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 5, 4, 3, 0, 1, 12, 11, 6, 4, 0, 1, 29, 28, 18, 8, 5, 0, 1, 70, 72, 48, 26, 10, 6, 0, 1, 169, 184, 130, 72, 35, 12, 7, 0, 1, 408, 469, 348, 204, 100, 45, 14, 8, 0, 1, 985, 1192, 927, 568, 295, 132, 56, 16, 9, 0, 1, 2378, 3022, 2456, 1571, 850, 404, 168, 68, 18, 10, 0, 1, 5741, 7644, 6477, 4312, 2430, 1200, 532, 208, 81, 20, 11, 0, 1

Offset: 0

Emanuele Munarini, Mar 26 2011

T(n,k) is the number of Dyck paths of height at most 3 with length 2n and k hills.
Row sum = F_(2n-1) Fibonacci number.
T is the convolution triangle of |A215936|. - Peter Luschny, Oct 19 2022

			Triangle begins:
1
0, 1
1, 0, 1
2, 2, 0, 1
5, 4, 3, 0, 1
12, 11, 6, 4, 0, 1
29, 28, 18, 8, 5, 0, 1
70, 72, 48, 26, 10, 6, 0, 1
169, 184, 130, 72, 35, 12, 7, 0, 1
408, 469, 348, 204, 100, 45, 14, 8, 0, 1

Vincenzo Librandi, Table of n, a(n) for n = 0..495

# Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
PMatrix(10, n -> (-1)^(n+1)*A215936(n)); # Peter Luschny, Oct 19 2022

Flatten[Table[Sum[Pochhammer[i,n-k-2i]/(n-k-2i)!Binomial[i+k,k]2^(n-k-2i),{i,0,(n-k)/2}],{n,0,12},{k,0,n}],1]

create_list(sum(pochhammer(i,n-k-2*i)/(n-k-2*i)!*binomial(i+k,k)*2^(n-k-2*i),i,0,(n-k)/2),n,0,12,k,0,n);

T(n,k) = sum(M(i,n-k-2i)*Binomial(i+k,k)*2^{n-k-2i},i=0..floor((n-k)/2)), where M(n,k)=n(n+1)(n+2)...(n+k-1)/k!.

Recurrence: T(n+2,k+1) = 2 T(n+1,k+1) + T(n+1,k) + T(n,k+1) - 2 T(n,k)

A202191 Triangle T(n,m) = coefficient of x^n in expansion of [x/(1-x-x^3)]^m = sum(n>=m, T(n,m) x^n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 3, 6, 6, 4, 1, 4, 11, 13, 10, 5, 1, 6, 18, 27, 24, 15, 6, 1, 9, 30, 51, 55, 40, 21, 7, 1, 13, 50, 94, 116, 100, 62, 28, 8, 1, 19, 81, 171, 234, 231, 168, 91, 36, 9, 1, 28, 130, 303, 460, 505, 420, 266, 128, 45, 10, 1, 41, 208

Offset: 1

Vladimir Kruchinin, Dec 14 2011

Convolution triangle of Narayana's cows sequence A000930. - Peter Luschny, Oct 09 2022

			Triangle T(n, m) starts:
[1]  1;
[2]  1,  1;
[3]  1,  2,  1;
[4]  2,  3,  3,   1;
[5]  3,  6,  6,   4,   1;
[6]  4, 11, 13,  10,   5,  1;
[7]  6, 18, 27,  24,  15,  6,  1;
[8]  9, 30, 51,  55,  40, 21,  7,  1;
[9] 13, 50, 94, 116, 100, 62, 28,  8,  1;
.
From _R. J. Mathar_, Mar 15 2013: (Start)
The matrix inverse starts
1;
-1,1;
1,-2,1;
-2,3,-3,1;
5,-6,6,-4,1;
-11,15,-13,10,-5,1;
24,-36,33,-24,15,-6,1;
-57,84,-84,63,-40,21,-7,1;
141,-204,208,-168,110,-62,28,-8,1.
(End)

Cf. A000930.

A202191 := proc(n,k)
    (x/(1-x-x^3))^k ;
    coeftayl(%,x=0,n) ;
end proc: # R. J. Mathar, Mar 15 2013
# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> simplify(hypergeom([(2 - n)/3, (3 - n)/3, (1 - n)/3], [(2 - n)/2, (1 - n)/2], -27/4))); # Peter Luschny, Oct 09 2022

T(n,m):=sum(binomial(k,(n-m-k)/2)*binomial(m+k-1,m-1)*((-1)^(n-m-k)+1),k,0,n-m)/2;

T(n,m)=sum(k=0..n-m, binomial(k,(n-m-k)/2)*binomial(m+k-1,m-1)*((-1)^(n-m-k)+1))/2.

A202710 Triangle read by rows. T(n, k) = coefficient of x^n in the Taylor expansion of [((1 - x - 2x^2 - sqrt(1 - 2x - 3x^2))/(2x^2))]^k.

Original entry on oeis.org

1, 2, 1, 4, 4, 1, 9, 12, 6, 1, 21, 34, 24, 8, 1, 51, 94, 83, 40, 10, 1, 127, 258, 267, 164, 60, 12, 1, 323, 707, 825, 604, 285, 84, 14, 1, 835, 1940, 2488, 2084, 1185, 454, 112, 16, 1, 2188, 5337, 7389, 6890, 4527, 2106, 679, 144, 18, 1, 5798, 14728, 21726, 22120, 16325, 8838, 3479, 968, 180, 20, 1

Offset: 1

Vladimir Kruchinin, Dec 23 2011

Triangle T(n,k)=
1. Riordan Array (1,((1-x-2*x^2-sqrt(1-2*x-3*x^2))/(2*x^2))) without first column.
2. Riordan Array (((1-x-2*x^2-sqrt(1-2*x-3*x^2))/(2*x)),((1-x-2*x^2-sqrt(1-2*x-3*x^2))/(2*x^2))) numbering triangle (0,0).
3. The leftmost column contains the Motzkin numbers A001006 without a(0).
The convolution triangle of the Motzkin numbers. - Peter Luschny, Oct 07 2022

			Triangle begins:
  1,
  2, 1,
  4, 4, 1,
  9, 12, 6, 1,
  21, 34, 24, 8, 1,
  51, 94, 83, 40, 10, 1,
  127, 258, 267, 164, 60, 12, 1

Cf. A001006.

# Uses function PMatrix from A357368. Adds a row and a column for n, k = 0.
PMatrix(10, n -> simplify(hypergeom([(1-n)/2, -n/2], [2], 4))); # Peter Luschny, Oct 06 2022

T[n_, k_] := Binomial[n - 1, n - k] + k*Sum[Binomial[n, i]*Binomial[k + i, n - k - i]/(k + i), {i, 0, n - k - 1}]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 06 2016, after Vladimir Kruchinin *)

T(n,k):=sum((i*(-1)^(k-i)*binomial(k,i)*sum(binomial(j+i,-n+2*j)*binomial(n+i,j+i) ,j,floor(n/2),n))/(n+i),i,1,k);

T(n,k):=+binomial(n-1,n-k)+k*sum((binomial(n,i)*binomial(k+i,n-k-i))/(k+i),i,0,n-k-1); /* Vladimir Kruchinin, Dec 06 2016*/

T(n,k) = Sum_{i=1..k} (i*(-1)^(k-i)*binomial(k,i)*Sum_{j=floor(n/2)..n} binomial(j+i,-n+2*j)*binomial(n+i,j+i))/(n+i).

T(n,k) = k*Sum_{i=0..n-k} binomial(k+i,n-k-i)*binomial(n,i)/(k+i). - Vladimir Kruchinin, Dec 09 2016

A206294 Riordan array (1, x/(1-x)^3).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 6, 6, 1, 0, 10, 21, 9, 1, 0, 15, 56, 45, 12, 1, 0, 21, 126, 165, 78, 15, 1, 0, 28, 252, 495, 364, 120, 18, 1, 0, 36, 462, 1287, 1365, 680, 171, 21, 1, 0, 45, 792, 3003, 4368, 3060, 1140, 231, 24, 1

Offset: 0

Philippe Deléham, Feb 05 2012

The convolution triangle of the triangular numbers A000217. - Peter Luschny, Oct 07 2022

			Triangle begins:
1
0, 1
0, 3, 1
0, 6, 6, 1
0, 10, 21, 9, 1
0, 15, 56, 45, 12, 1
0, 21, 126, 165, 78, 15, 1
0, 28, 252, 495, 364, 120, 18, 1
0, 36, 462, 1287, 1365, 680, 171, 21, 1
0, 45, 792, 3003, 4368, 3060, 1140, 231, 24, 1
0, 55, 1287, 6435, 12376, 11628, 5985, 1771, 300, 27, 1
0, 66, 2002, 12870, 31824, 38760, 26324, 10626, 2600, 378, 30, 1

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. Columns: A000007, A000217 (triangular numbers), A000389, A000581, A001288, A010967..(+3)..A011000, A017714..(+3)..A017762.
Row sums are A052529.
Cf. A127893.

# Uses function PMatrix from A357368.
PMatrix(10, n -> n * (n + 1) / 2); # Peter Luschny, Oct 07 2022

Table[If[n == 0 && k == 0 , 1, Binomial[n - 1 + 2 k, n - k]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 25 2017 *)

{T(n,k)=polcoeff(1/(1-x+x*O(x^(n-k)))^(3*k),n-k)}

{T(n,k)=polcoeff(polcoeff((1-x)^3/((1-x)^3-y*x +x*O(x^n)),n,x),k,y)}
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))

Triangle T(n,k), read by rows, given by (0, 3, -1, 2/3, -1/6, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
T(n,0) = 0^n, T(n,k) = C(n-1+2k, n-k) for k > 0.
T(n,n) = 1, T(k+1,k) = 3*k = A008585(k), T(k+2,k) = A081266(k).
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A052529(n), A052910(n) for x = 0, 1, 2 respectively.
G.f.: (1-x)^3/((1-x)^3-y*x).

cp's OEIS Frontend

A075435 T(n,k) = right- or upward-moving paths connecting opposite corners of an n X n chessboard, visiting the diagonal at k points between start and finish.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Maxima

Sage

Formula

A105495 Triangle read by rows: T(n,k) is the number of compositions of n into k parts when parts equal to q are of q^2 kinds.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Maxima

SageMath

Formula

A127898 Inverse of Riordan array (1/(1+x)^3, x/(1+x)^3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

A206306 Riordan array (1, x/(1-3*x+2*x^2)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A357583 Triangle read by rows. Convolution triangle of the Bell numbers.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

A128627 Triangle read by rows. Convolution triangle based on A002865.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Extensions

A188285 Riordan matrix ( (1-2x)/(1-2x-x^2), (x-2x^2)/(1-2x-x^2) ).

Views

Author

Keywords

Comments

A206306 Riordan array (1, x/(1-3x+2x^2)).

A202710 Triangle read by rows. T(n, k) = coefficient of x^n in the Taylor expansion of [((1 - x - 2x^2 - sqrt(1 - 2x - 3x^2))/(2x^2))]^k.