A023531 -id:A023531 - cp's OEIS frontend

A291204 Number F(n,h,t) of forests of t labeled rooted trees with n vertices such that the root of each subtree contains the subtree's minimal label and h is the maximum of 0 and the tree heights; triangle of triangles F(n,h,t), n>=0, h=0..n, t=0..n-h, read by layers, then by rows.

1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 3, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 7, 6, 0, 4, 4, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 15, 25, 10, 0, 14, 30, 10, 0, 8, 5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 31, 90, 65, 15, 0, 51, 174, 120, 20, 0, 54, 63, 15, 0, 13, 6, 0, 1, 0

Offset: 0

Alois P. Heinz, Aug 20 2017

Elements in rows h=0 give A023531.
Positive elements in rows h=1 give A008277.
Positive row sums per layer (and - with a different offset - positive elements in column t=1) give A179454.
Positive column sums per layer give A132393.

			n h\t: 0  1  2  3  4 5 : A179454 : A132393       : A000142
-----+-----------------+---------+---------------+--------
0 0  : 1               :       1 :  1            : 1
-----+-----------------+---------+---------------+--------
1 0  : 0  1            :       1 :  .            :
1 1  : 0               :         :  1            : 1
-----+-----------------+---------+---------------+--------
2 0  : 0  0  1         :       1 :  .  .         :
2 1  : 0  1            :       1 :  .            :
2 2  : 0               :         :  1  1         : 2
-----+-----------------+---------+---------------+--------
3 0  : 0  0  0  1      :       1 :  .  .  .      :
3 1  : 0  1  3         :       4 :  .  .         :
3 2  : 0  1            :       1 :  .            :
3 3  : 0               :         :  2  3  1      : 6
-----+-----------------+---------+---------------+--------
4 0  : 0  0  0  0  1   :       1 :  .  .  .  .   :
4 1  : 0  1  7  6      :      14 :  .  .  .      :
4 2  : 0  4  4         :       8 :  .  .         :
4 3  : 0  1            :       1 :  .            :
4 4  : 0               :         :  6 11  6  1   : 24
-----+-----------------+---------+---------------+--------
5 0  : 0  0  0  0  0 1 :       1 :  .  .  .  . . :
5 1  : 0  1 15 25 10   :      51 :  .  .  .  .   :
5 2  : 0 14 30 10      :      54 :  .  .  .      :
5 3  : 0  8  5         :      13 :  .  .         :
5 4  : 0  1            :       1 :  .            :
5 5  : 0               :         : 24 50 35 10 1 : 120
-----+-----------------+---------+---------------+--------

Alois P. Heinz, Layers n = 0..48, flattened

Cf. A000007, A000012, A000110, A000142, A000217, A000225, A000254, A000325, A001791, A007820, A008277, A023531, A048993, A057427, A058692, A179454, A291203, A291336, A291529.

b:= proc(n, t, h) option remember; expand(`if`(n=0 or h=0, x^(t*n), add(
       binomial(n-1, j-1)*x^t*b(j-1, 0, h-1)*b(n-j, t, h), j=1..n)))
    end:
g:= (n, h)-> b(n, 1, h)-`if`(h=0, 0, b(n, 1, h-1)):
F:= (n, h, t)-> coeff(g(n, h), x, t):
seq(seq(seq(F(n, h, t), t=0..n-h), h=0..n), n=0..8);

b[n_, t_, h_] := b[n, t, h] = Expand[If[n == 0 || h == 0, x^(t*n), Sum[Binomial[n-1, j-1]*x^t*b[j-1, 0, h-1]*b[n-j, t, h], {j, 1, n}]]];
g[n_, h_] := b[n, 1, h] - If[h == 0, 0, b[n, 1, h - 1]];
F[n_, h_, t_] := Coefficient[g[n, h], x, t];
Table[Table[Table[F[n, h, t], {t, 0, n - h}], {h, 0, n}], {n, 0, 8}] // Flatten (* Jean-François Alcover, Mar 17 2022, after Alois P. Heinz *)

Sum_{i=0..n} F(n,i,n-i) = A000325(n).
Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A000142(n).
Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A000254(n).
Sum_{t=0..n-1} F(n,1,t) = A058692(n) =  A000110(n) - 1.
F(2n,n,n) = A001791(n) for n>0.
F(2n,1,n) = A007820(n).
F(n,1,n-1) = A000217(n-1) for n>0.
F(n,n-1,1) = A057427(n).
F(n,1,2) = A000225(n-1) for n>2.
F(n,0,n) = 1 = A000012(n).
F(n,0,0) = A000007(n).

A291336 Number F(n,h,t) of forests of t unlabeled rooted trees with n vertices such that h is the maximum of 0 and the tree heights; triangle of triangles F(n,h,t), n>=0, h=0..n, t=0..n-h, read by layers, then by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 2, 1, 0, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 2, 1, 0, 4, 3, 1, 0, 3, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 3, 3, 2, 1, 0, 6, 8, 3, 1, 0, 8, 4, 1, 0, 4, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 3, 4, 3, 2, 1, 0, 10, 15, 9, 3, 1, 0, 18, 13, 4, 1, 0, 13, 5, 1, 0, 5, 1, 0, 1, 0

Offset: 0

Alois P. Heinz, Aug 22 2017

Elements in rows h=0 give A023531.
Positive elements in rows h=1 give A008284.
Positive row sums per layer (and - with a different offset - positive elements in column t=1) give A034781.
Positive column sums per layer give A033185.

			n h\t: 0 1 2 3 4 5 : A034781 : A033185   : A000081
-----+-------------+---------+-----------+--------
0 0  : 1           :         :           : 1
-----+-------------+---------+-----------+--------
1 0  : 0 1         :       1 : .         :
1 1  : 0           :         : 1         : 1
-----+-------------+---------+-----------+--------
2 0  : 0 0 1       :       1 : . .       :
2 1  : 0 1         :       1 : .         :
2 2  : 0           :         : 1 1       : 2
-----+-------------+---------+-----------+--------
3 0  : 0 0 0 1     :       1 : . . .     :
3 1  : 0 1 1       :       2 : . .       :
3 2  : 0 1         :       1 : .         :
3 3  : 0           :         : 2 1 1     : 4
-----+-------------+---------+-----------+--------
4 0  : 0 0 0 0 1   :       1 : . . . .   :
4 1  : 0 1 2 1     :       4 : . . .     :
4 2  : 0 2 1       :       3 : . .       :
4 3  : 0 1         :       1 : .         :
4 4  : 0           :         : 4 3 1 1   : 9
-----+-------------+---------+-----------+--------
5 0  : 0 0 0 0 0 1 :       1 : . . . . . :
5 1  : 0 1 2 2 1   :       6 : . . . .   :
5 2  : 0 4 3 1     :       8 : . . .     :
5 3  : 0 3 1       :       4 : . .       :
5 4  : 0 1         :       1 : .         :
5 5  : 0           :         : 9 6 3 1 1 : 20
-----+-------------+---------+-----------+--------

Alois P. Heinz, Layers n = 0..48, flattened

Cf. A000007, A000041, A000065, A000081, A001853, A005197, A008284, A023531, A033185, A034781, A291203, A291204, A291529.

b:= proc(n, i, t, h) option remember; expand(`if`(n=0 or h=0
       or i=1, x^(t*n), b(n, i-1, t, h)+add(x^(t*j)*binomial(
       b(i-1$2, 0, h-1)+j-1, j)*b(n-i*j, i-1, t, h), j=1..n/i)))
    end:
g:= (n, h)-> b(n$2, 1, h)-`if`(h=0, 0, b(n$2, 1, h-1)):
F:= (n, h, t)-> coeff(g(n, h), x, t):
seq(seq(seq(F(n, h, t), t=0..n-h), h=0..n), n=0..9);

b[n_, i_, t_, h_] := b[n, i, t, h] = Expand[If[n == 0 || h == 0
     || i == 1, x^(t*n), b[n, i-1, t, h] + Sum[x^(t*j)*Binomial[
     b[i-1, i-1, 0, h-1]+j-1, j]*b[n - i*j, i-1, t, h], {j, 1, n/i}]]];
g[n_, h_] := b[n, n, 1, h] - If[h == 0, 0, b[n, n, 1, h-1]];
F[n_, h_, t_] := Coefficient[g[n, h], x, t];
Table[Table[Table[F[n, h, t], {t, 0, n-h}], {h, 0, n}], {n, 0, 9}] //
Flatten (* Jean-François Alcover, Mar 10 2022, after Alois P. Heinz *)

Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A000081(n+1).
Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A005197(n).
Sum_{h=0..n} Sum_{t=0..n-h} (h+1) * F(n,h,t) = A001853(n+1) for n>0.
Sum_{t=0..n-1} F(n,1,t) = A000065(n) = A000041(n) - 1.
F(n,1,1) = 1 for n>1.
F(n,0,0) = A000007(n).

A339494 T(n, k) is the number of domino towers of n bricks with height at most 3 and k bricks in the base floor. Triangle read by rows, T(n, k) for 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 5, 9, 4, 1, 3, 14, 14, 5, 1, 1, 16, 29, 20, 6, 1, 0, 12, 46, 51, 27, 7, 1, 0, 5, 52, 101, 81, 35, 8, 1, 0, 1, 41, 150, 190, 120, 44, 9, 1, 0, 0, 22, 169, 345, 323, 169, 54, 10, 1, 0, 0, 7, 143, 495, 687, 511, 229, 65, 11, 1

Offset: 1

Peter Luschny, Dec 07 2020

This is the third triangle in a sequence of triangles: The first is the unit triangle A023531; the second is the binomial triangle C(k, n-k) without the first column, triangle A030528. This triangle highlights the connection between the Pascal triangle and the Fibonacci numbers in the case m = 2. Similarly, the current triangle and its row sums generalizes this to the case m = 3 of the construction of Union(A333650(n, j), j=1..m), classified by the number of bricks in the base floor.

			Triangle starts:        n: [row] sum
                          1: [1] 1
                         2: [2, 1] 3
                       3: [5, 3, 1] 9
                     4: [5, 9, 4, 1] 19
                   5: [3, 14, 14, 5, 1] 37
                 6: [1, 16, 29, 20, 6, 1] 73
              7: [0, 12, 46, 51, 27, 7, 1] 144
            8: [0, 5, 52, 101, 81, 35, 8, 1] 283
         9: [0, 1, 41, 150, 190, 120, 44, 9, 1] 556
     10: [0, 0, 22, 169, 345, 323, 169, 54, 10, 1] 1093

Peter Luschny, Domino towers classified by the size of the base floor. Illustrating row 4 of the triangle.

Cf. A339495 (row sums), A333650, A030528, A023531.

A049325 A convolution triangle of numbers generalizing Pascal's triangle A007318.

Original entry on oeis.org

1, 6, 1, 16, 12, 1, 16, 68, 18, 1, 0, 224, 156, 24, 1, 0, 448, 840, 280, 30, 1, 0, 512, 3072, 2080, 440, 36, 1, 0, 256, 7872, 10896, 4160, 636, 42, 1, 0, 0, 14080, 42240, 28240, 7296, 868, 48, 1, 0, 0, 16896, 123904, 145376, 60720, 11704, 1136, 54, 1, 0, 0, 12288

Offset: 1

Wolfdieter Lang

			{1}; {6,1}; {16,12,1}; {16,68,18,1}; {0,224,156,24,1}; ...

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

a(n, m) := s1(-3, n, m), a member of a sequence of triangles including s1(0, n, m)= A023531(n, m) (unit matrix) and s1(2, n, m)=A007318(n-1, m-1) (Pascal's triangle). s1(-1, n, m)= A030528, s1(-2, n, m)= A049324(n, m).

Cf. A049349.

a(n, m) = 4*(4*m-n+1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, nA033842(3, m)).

A049326 A convolution triangle of numbers generalizing Pascal's triangle A007318.

Original entry on oeis.org

1, 10, 1, 50, 20, 1, 125, 200, 30, 1, 125, 1250, 450, 40, 1, 0, 5250, 4375, 800, 50, 1, 0, 15000, 30375, 10500, 1250, 60, 1, 0, 28125, 157500, 100500, 20625, 1800, 70, 1, 0, 31250, 621875, 740000, 250625, 35750, 2450, 80, 1, 0, 15625, 1875000, 4318750

Offset: 1

Wolfdieter Lang

			{1}; {10,1}; {50,20,1}; {125,200,30,1}; {125,1250,450,40,1}; ...

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

a(n, m) := s1(-4, n, m), a member of a sequence of triangles including s1(0, n, m)= A023531(n, m) (unit matrix) and s1(2, n, m)=A007318(n-1, m-1) (Pascal's triangle). s1(-1, n, m)= A030528.

Cf. A049350.

a(n, m) = 5*(5*m-n+1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, nA033842(4, m)).

A049327 A convolution triangle of numbers generalizing Pascal's triangle A007318.

Original entry on oeis.org

1, 15, 1, 120, 30, 1, 540, 465, 45, 1, 1296, 4680, 1035, 60, 1, 1296, 33192, 15795, 1830, 75, 1, 0, 171072, 176688, 37260, 2850, 90, 1, 0, 641520, 1521828, 563409, 72450, 4095, 105, 1, 0, 1710720, 10359360, 6686064, 1375605, 124740, 5565, 120, 1

Offset: 1

Wolfdieter Lang

			{1}; {15,1}; {120,30,1}; {540,465,45,1}; {1296,4680,1035,60,1}; ...

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

a(n, m) := s1(-5, n, m), a member of a sequence of triangles including s1(0, n, m)= A023531(n, m) (unit matrix) and s1(2, n, m)=A007318(n-1, m-1) (Pascal's triangle). s1(-1, n, m)= A030528.

Cf. A049351.

a(n, m) = 6*(6*m-n+1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, nA033842(5, m)).

A194704 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (4 + m).

Original entry on oeis.org

5, 1, 4, 1, 2, 2, 0, 1, 1, 3, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2

Offset: 1

Omar E. Pol, Feb 05 2012

Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 4. For further information see A182703 and A135010.

			Triangle begins:
  5,
  1, 4,
  1, 2, 2,
  0, 1, 1, 3,
  1, 0, 1, 1, 2,
  ...
For k = 1 and m = 1: T(1,1) = 5 because there are five parts of size 1 in the last section of the set of partitions of 5, since 4 + m = 5, so a(1) = 5.
For k = 2 and m = 1: T(2,1) = 1 because there is only one part of size 2 in the last section of the set of partitions of 5, since 4 + m = 5, so a(2) = 1.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Always the sum of row k = p(4) = A000041(4) = 5.
The first (0-10) members of this family of triangles are A023531, A129186, A194702, A194703, this sequence, A194705-A194710.
Cf. A135010, A138121, A182712-A182714, A194812.

P(n)={my(M=matrix(n,n), d=4); M[1,1]=numbpart(d); for(m=1, n, forpart(p=m+d, for(k=1, #p, my(t=p[k]); if(t<=n && m<=t, M[t, m]++)), [2, m+d])); M}
{ my(T=P(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Feb 19 2020

T(k,m) = A182703(4+m,k), with T(k,m) = 0 if k > 4+m.

T(k,m) = A194812(4+m,k).

Terms a(16) and beyond from Andrew Howroyd, Feb 19 2020

A194705 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (5 + m).

Original entry on oeis.org

7, 4, 3, 2, 2, 3, 1, 1, 3, 2, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 2, 0, 1, 0, 1, 1, 2, 2, 0, 0, 1, 0, 1, 1, 2, 2, 0, 0, 0, 1, 0, 1, 1, 2, 2, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2

Offset: 1

Omar E. Pol, Feb 05 2012

Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 5. For further information see A182703 and A135010.

			Triangle begins:
  7,
  4, 3,
  2, 2, 3,
  1, 1, 3, 2,
  0, 1, 1, 2, 3,
  1, 0, 1, 1, 2, 2,
  0, 1, 0, 1, 1, 2, 2,
  ...
For k = 1 and m = 1: T(1,1) = 7 because there are seven parts of size 1 in the last section of the set of partitions of 6, since 5 + m = 6, so a(1) = 7.
For k = 2 and m = 1: T(2,1) = 4 because there are four parts of size 2 in the last section of the set of partitions of 6, since 5 + m = 6, so a(2) = 4.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Always the sum of row k = p(5) = A000041(5) = 7.
The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194704, this sequence, A194706-A194710.
Cf. A135010, A138121, A194812.

P(n)={my(M=matrix(n,n), d=5); M[1,1]=numbpart(d); for(m=1, n, forpart(p=m+d, for(k=1, #p, my(t=p[k]); if(t<=n && m<=t, M[t, m]++)), [2, m+d])); M}
{ my(T=P(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Feb 19 2020

T(k,m) = A182703(5+m,k), with T(k,m) = 0 if k > 5+m.

T(k,m) = A194812(5+m,k).

Terms a(29) and beyond from Andrew Howroyd, Feb 19 2020

A194706 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (6 + m).

Original entry on oeis.org

11, 3, 8, 2, 3, 6, 1, 3, 2, 5, 1, 1, 2, 3, 4, 0, 1, 1, 2, 2, 5, 1, 0, 1, 1, 2, 2, 4, 0, 1, 0, 1, 1, 2, 2, 4, 0, 0, 1, 0, 1, 1, 2, 2, 4, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4

Offset: 1

Omar E. Pol, Feb 05 2012

Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 6. For further information see A182703 and A135010.

			Triangle begins:
  11,
   3, 8,
   2, 3, 6,
   1, 3, 2, 5,
   1, 1, 2, 3, 4,
   0, 1, 1, 2, 2, 5,
  ...
For k = 1 and m = 1: T(1,1) = 11 because there are 11 parts of size 1 in the last section of the set of partitions of 7, since 6 + m = 7, so a(1) = 11.
For k = 2 and m = 1: T(2,1) = 3 because there are three parts of size 2 in the last section of the set of partitions of 7, since 6 + m = 7, so a(2) = 3.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Always the sum of row k = p(6) = A000041(6) = 11.
The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194705, this sequence, A194707-A194710.
Cf. A135010, A138121, A194812.

P(n)={my(M=matrix(n,n), d=6); M[1,1]=numbpart(d); for(m=1, n, forpart(p=m+d, for(k=1, #p, my(t=p[k]); if(t<=n && m<=t, M[t, m]++)), [2, m+d])); M}
{ my(T=P(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Feb 19 2020

T(k,m) = A182703(6+m,k), with T(k,m) = 0 if k > 6+m.

T(k,m) = A194812(6+m,k).

Terms a(22) and beyond from Andrew Howroyd, Feb 19 2020

A194707 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (7 + m).

Original entry on oeis.org

15, 8, 7, 3, 6, 6, 3, 2, 5, 5, 1, 2, 3, 4, 5, 1, 1, 2, 2, 5, 4, 0, 1, 1, 2, 2, 4, 5, 1, 0, 1, 1, 2, 2, 4, 4, 0, 1, 0, 1, 1, 2, 2, 4, 4, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4

Offset: 1

Omar E. Pol, Feb 05 2012

Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 7. For further information see A182703 and A135010.

			Triangle begins:
  15,
   8, 7,
   3, 6, 6,
   3, 2, 5, 5,
  ...
For k = 1 and m = 1: T(1,1) = 15 because there are 15 parts of size 1 in the last section of the set of partitions of 8, since 7 + m = 8, so a(1) = 15.
For k = 2 and m = 1: T(2,1) = 8 because there are eight parts of size 2 in the last section of the set of partitions of 8, since 7 + m = 8, so a(2) = 8.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Always the sum of row k = p(7) = A000041(7) = 15.
The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194706, this sequence, A194708-A194710.
Cf. A135010, A138121, A194812.

P(n)={my(M=matrix(n,n), d=7); M[1,1]=numbpart(d); for(m=1, n, forpart(p=m+d, for(k=1, #p, my(t=p[k]); if(t<=n && m<=t, M[t, m]++)), [2, m+d])); M}
{ my(T=P(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Feb 19 2020

T(k,m) = A182703(7+m,k), with T(k,m) = 0 if k > 7+m.

T(k,m) = A194812(7+m,k).

Terms a(11) and beyond from Andrew Howroyd, Feb 19 2020

cp's OEIS Frontend

A291204 Number F(n,h,t) of forests of t labeled rooted trees with n vertices such that the root of each subtree contains the subtree's minimal label and h is the maximum of 0 and the tree heights; triangle of triangles F(n,h,t), n>=0, h=0..n, t=0..n-h, read by layers, then by rows.

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A291336 Number F(n,h,t) of forests of t unlabeled rooted trees with n vertices such that h is the maximum of 0 and the tree heights; triangle of triangles F(n,h,t), n>=0, h=0..n, t=0..n-h, read by layers, then by rows.

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A339494 T(n, k) is the number of domino towers of n bricks with height at most 3 and k bricks in the base floor. Triangle read by rows, T(n, k) for 1 <= k <= n.

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A049325 A convolution triangle of numbers generalizing Pascal's triangle A007318.

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A049326 A convolution triangle of numbers generalizing Pascal's triangle A007318.

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A049327 A convolution triangle of numbers generalizing Pascal's triangle A007318.

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A194704 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (4 + m).

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A194705 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (5 + m).

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