cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A291529 Number F(n,h,t) of forests of t (unlabeled) rooted identity 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, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 3, 0, 0, 3, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 5, 1, 0, 0, 5, 4, 0, 0, 4, 1, 0, 1, 0
Offset: 0

Views

Author

Alois P. Heinz, Aug 25 2017

Keywords

Comments

Positive row sums per layer (and - with a different offset - positive elements in column t=1) give A227819.
Positive column sums per layer give A227774.

Examples

			n h\t: 0 1 2 3 4 5 : A227819 : A227774   : A004111
-----+-------------+---------+-----------+--------
0 0  : 1           :         :           : 1
-----+-------------+---------+-----------+--------
1 0  : 0 1         :       1 : .         :
1 1  : 0           :         : 1         : 1
-----+-------------+---------+-----------+--------
2 0  : 0 0 0       :       0 : . .       :
2 1  : 0 1         :       1 : .         :
2 2  : 0           :         : 1 0       : 1
-----+-------------+---------+-----------+--------
3 0  : 0 0 0 0     :       0 : . . .     :
3 1  : 0 0 1       :       1 : . .       :
3 2  : 0 1         :       1 : .         :
3 3  : 0           :         : 1 1 0     : 2
-----+-------------+---------+-----------+--------
4 0  : 0 0 0 0 0   :       0 : . . . .   :
4 1  : 0 0 0 0     :       0 : . . .     :
4 2  : 0 1 1       :       2 : . .       :
4 3  : 0 1         :       1 : .         :
4 4  : 0           :         : 2 1 0 0   : 3
-----+-------------+---------+-----------+--------
5 0  : 0 0 0 0 0 0 :       0 : . . . . . :
5 1  : 0 0 0 0 0   :       0 : . . . .   :
5 2  : 0 0 2 0     :       2 : . . .     :
5 3  : 0 2 1       :       3 : . .       :
5 4  : 0 1         :       1 : .         :
5 5  : 0           :         : 3 3 0 0 0 : 6
-----+-------------+---------+-----------+--------

Links

Crossrefs

Cf. A000007, A004111, A227774, A227819, A291203, A291204, A291336, A291532, A291559.

Programs

  • Maple
    b:= proc(n, i, t, h) option remember; expand(`if`(n=0 or h=0 or i=1,
      `if`(n<2, x^(t*n), 0), b(n, i-1, t, h)+add(x^(t*j)*binomial(
       b(i-1$2, 0, h-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..10);
  • Mathematica
    b[n_, i_, t_, h_] := b[n, i, t, h] = Expand[If[n == 0 || h == 0 || i == 1, If[n < 2, x^(t*n), 0], b[n, i - 1, t, h] + Sum[x^(t*j)*Binomial[b[i - 1, i - 1, 0, h - 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[F[n, h, t], {n, 0, 10}, {h, 0, n}, {t, 0, n - h}] // Flatten (* Jean-François Alcover, Jun 04 2018, from Maple *)

Formula

Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A004111(n+1).
Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A291532(n).
Sum_{h=0..n-2} Sum_{t=1..n-1-h} (h+1) * F(n-1,h,t) = A291559(n).
F(n,0,0) = A000007(n).

A291203 Number F(n,h,t) of forests of t labeled 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, 2, 0, 0, 0, 0, 1, 0, 3, 6, 0, 6, 0, 0, 0, 0, 0, 1, 0, 4, 24, 12, 0, 36, 24, 0, 24, 0, 0, 0, 0, 0, 0, 1, 0, 5, 80, 90, 20, 0, 200, 300, 60, 0, 300, 120, 0, 120, 0, 0, 0, 0, 0, 0, 0, 1, 0, 6, 240, 540, 240, 30, 0, 1170, 3000, 1260, 120, 0, 3360, 2520, 360, 0, 2520, 720, 0, 720, 0
Offset: 0

Views

Author

Alois P. Heinz, Aug 20 2017

Keywords

Comments

Positive elements in column t=1 give A034855.
Elements in rows h=0 give A023531.
Elements in rows h=1 give A059297.
Positive row sums per layer give A235595.
Positive column sums per layer give A061356.

Examples

			n h\t: 0   1   2  3  4 5 : A235595 : A061356          : A000272
-----+-------------------+---------+------------------+--------
0 0  : 1                 :         :                  : 1
-----+-------------------+---------+------------------+--------
1 0  : 0   1             :      1  :   .              :
1 1  : 0                 :         :   1              : 1
-----+-------------------+---------+------------------+--------
2 0  : 0   0   1         :      1  :   .   .          :
2 1  : 0   2             :      2  :   .              :
2 2  : 0                 :         :   2   1          : 3
-----+-------------------+---------+------------------+--------
3 0  : 0   0   0  1      :      1  :   .   .   .      :
3 1  : 0   3   6         :      9  :   .   .          :
3 2  : 0   6             :      6  :   .              :
3 3  : 0                 :         :   9   6   1      : 16
-----+-------------------+---------+------------------+--------
4 0  : 0   0   0  0  1   :      1  :   .   .   .  .   :
4 1  : 0   4  24 12      :     40  :   .   .   .      :
4 2  : 0  36  24         :     60  :   .   .          :
4 3  : 0  24             :     24  :   .              :
4 4  : 0                 :         :  64  48  12  1   : 125
-----+-------------------+---------+------------------+--------
5 0  : 0   0   0  0  0 1 :      1  :   .   .   .  . . :
5 1  : 0   5  80 90 20   :    195  :   .   .   .  .   :
5 2  : 0 200 300 60      :    560  :   .   .   .      :
5 3  : 0 300 120         :    420  :   .   .          :
5 4  : 0 120             :    120  :   .              :
5 5  : 0                 :         : 625 500 150 20 1 : 1296
-----+-------------------+---------+------------------+--------

Links

Crossrefs

Cf. A000007, A000012, A000142, A000272, A000551, A001477, A001788, A001854, A002378, A023531, A034855, A038154, A059297, A061356, A089946, A126804, A234953, A235595, A235596, A243014, A291204, A291336, A291529.

Programs

  • Maple
    b:= proc(n, t, h) option remember; expand(`if`(n=0 or h=0, x^(t*n), add(
       binomial(n-1, j-1)*j*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);
  • Mathematica
    b[n_, t_, h_] := b[n, t, h] = Expand[If[n == 0 || h == 0, x^(t*n), Sum[
     Binomial[n-1, j-1]*j*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 *)

Formula

Sum_{i=0..n} F(n,i,n-i) = A243014(n) = 1 + A038154(n).
Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A000272(n+1).
Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A089946(n-1) for n>0.
Sum_{h=0..n} Sum_{t=0..n-h} (h+1) * F(n,h,t) = A234953(n+1) for n>0.
Sum_{h=0..n} Sum_{t=0..n-h} (h+1)*(n+1) * F(n,h,t) = A001854(n+1) for n>0.
Sum_{t=0..n-1} F(n,1,t) = A235596(n+1).
F(2n,n,n) = A126804(n) for n>0.
F(n,0,n) = 1 = A000012(n).
F(n,1,1) = n = A001477(n) for n>1.
F(n,n-1,1) = n! = A000142(n) for n>0.
F(n,1,n-1) = A002378(n-1) for n>0.
F(n,2,1) = A000551(n).
F(n,3,1) = A000552(n).
F(n,4,1) = A000553(n).
F(n,1,2) = A001788(n-1) for n>2.
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

Views

Author

Alois P. Heinz, Aug 22 2017

Keywords

Comments

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.

Examples

			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
-----+-------------+---------+-----------+--------

Links

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

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).
Showing 1-3 of 3 results.

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