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.

A095133 Triangle of numbers of forests on n nodes containing k trees.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 1, 6, 6, 4, 2, 1, 1, 11, 11, 7, 4, 2, 1, 1, 23, 23, 14, 8, 4, 2, 1, 1, 47, 46, 29, 15, 8, 4, 2, 1, 1, 106, 99, 60, 32, 16, 8, 4, 2, 1, 1, 235, 216, 128, 66, 33, 16, 8, 4, 2, 1, 1, 551, 488, 284, 143, 69, 34, 16, 8, 4, 2, 1, 1, 1301, 1121, 636, 315, 149, 70, 34, 16, 8, 4, 2, 1, 1
Offset: 1

Views

Author

Eric W. Weisstein, May 29 2004

Keywords

Comments

Row sums are A005195.
For k > n/2, T(n,k) = T(n-1,k-1). - Geoffrey Critzer, Oct 13 2012

Examples

			Triangle begins:
    1;
    1,  1;
    1,  1,  1;
    2,  2,  1,  1;
    3,  3,  2,  1,  1;
    6,  6,  4,  2,  1, 1;
   11, 11,  7,  4,  2, 1, 1;
   23, 23, 14,  8,  4, 2, 1, 1;
   47, 46, 29, 15,  8, 4, 2, 1, 1;
  106, 99, 60, 32, 16, 8, 4, 2, 1, 1;
  ...

Links

Crossrefs

Cf. A005195 (row sums), A005196, A106240, A000055 (first column), A274937 (2nd column), A105821.
Limiting sequence of reversed rows gives A215930.
Reflected table is A136605. - Alois P. Heinz, Apr 11 2014

Programs

  • Maple
    with(numtheory):
b:= proc(n) option remember; local d, j; `if` (n<=1, n,
      (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/(n-1))
    end:
t:= proc(n) option remember; local k; `if` (n=0, 1,
      b(n)-(add(b(k)*b(n-k), k=0..n)-`if`(irem(n, 2)=0, b(n/2), 0))/2)
    end:
g:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(g(n-i*j, i-1, p-j) *
       binomial(t(i)+j-1, j), j=0..min(n/i, p)))))
    end:
a:= (n, k)-> g(n, n, k):
seq(seq(a(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Aug 20 2012
  • Mathematica
    nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i]s[n-1,i]i,{i,1,n-1}]/(n-1);ft=Table[a[i]-Sum[a[j]a[i-j],{j,1,i/2}]+If[OddQ[i],0,a[i/2](a[i/2]+1)/2],{i,1,nn}];CoefficientList[Series[Product[1/(1-y x^i)^ft[[i]],{i,1,nn}],{x,0,20}],{x,y}]//Grid (* Geoffrey Critzer, Oct 13 2012, after code given by Robert A. Russell in A000055 *)

Formula

T(n, k) = sum over the partitions of n, 1M1 + 2M2 + ... + nMn, with exactly k parts, of Product_{i=1..n} binomial(A000055(i) + Mi - 1, Mi). - Washington Bomfim, May 12 2005

Extensions

More terms from Vladeta Jovovic, Jun 03 2004

A215930 Number of forests on unlabeled nodes with n edges and no single node trees.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 34, 71, 154, 341, 768, 1765, 4134, 9838, 23766, 58226, 144353, 361899, 916152, 2339912, 6023447, 15617254, 40752401, 106967331, 282267774, 748500921, 1993727506, 5332497586, 14316894271, 38574473086, 104273776038, 282733466684, 768809041078
Offset: 0

Views

Author

Alois P. Heinz, Aug 27 2012

Keywords

Comments

Each forest counted by a(n) with n>0 has number of nodes from the interval [n+1,2*n] and number of trees in [1,n].
Also limiting sequence of reversed rows of A095133.
Differs from A011782 first at n=6 (32) and from A088325 at n=8 (153).

Examples

			a(0) = 1: (  ), the empty forest with 0 trees and 0 edges.
a(1) = 1: ( o-o ), 1 tree and 1 edge.                      o
a(2) = 2: ( o-o-o ), ( o-o o-o ).                          |
a(3) = 4: ( o-o-o-o ), ( o-o-o o-o ), ( o-o o-o o-o ), ( o-o-o ).

Links

Crossrefs

Cf. A000055, A005195, A011782, A051491, A088325, A095133, A105821, A136605.

Programs

  • Maple
    with(numtheory):
b:= proc(n) option remember; local d, j; `if`(n<=1, n,
      (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/(n-1))
    end:
t:= proc(n) option remember; local k; `if` (n=0, 1, b(n)-
      (add(b(k)*b(n-k), k=0..n)-`if`(irem(n, 2)=0, b(n/2), 0))/2)
    end:
g:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(g(n-i*j, i-1, p-j)*
       binomial(t(i)+j-1, j), j=0..min(n/i, p)))))
    end:
a:= n-> g(2*n, 2*n, n):
seq(a(n), n=0..40);
  • Mathematica
    nn = 30; t[x_] := Sum[a[n] x^n, {n, 1, nn}]; a[0] = 0;
a[1] = 1; sol =
SolveAlways[
  0 == Series[
    t[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}], x];
b[x_] := Sum[a[n] x^n /. sol, {n, 0, nn}]; ft =
Drop[Flatten[
   CoefficientList[Series[b[x] - (b[x]^2 - b[x^2])/2, {x, 0, nn}],
    x]], 1]; Drop[
CoefficientList[
  Series[Product[1/(1 - y ^(i - 1))^ft[[i]], {i, 2, nn}], {y, 0, nn}],
y], -1] (* Geoffrey Critzer, Nov 10 2014 *)

Formula

a(n) = A095133(2*n,n).
a(n) = A105821(2*n+1,n+1). - Alois P. Heinz, Jul 10 2013
a(n) = A136605(2*n+1,n). - Alois P. Heinz, Apr 11 2014
a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.955765285..., c = 3.36695186... . - Vaclav Kotesovec, Sep 10 2014

A105820 Triangle giving the numbers of different forests of m trees of smallest order 2, i.e., without isolated vertices.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 3, 1, 0, 0, 0, 6, 3, 1, 0, 0, 0, 11, 5, 1, 0, 0, 0, 0, 23, 12, 3, 1, 0, 0, 0, 0, 47, 23, 6, 1, 0, 0, 0, 0, 0, 106, 52, 14, 3, 1, 0, 0, 0, 0, 0, 235, 110, 29, 6, 1, 0, 0, 0, 0, 0, 0, 551, 253, 68, 15, 3, 1, 0, 0, 0, 0, 0, 0, 1301, 570, 148, 31, 6, 1, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Washington Bomfim, Apr 25 2005

Keywords

Comments

Forests of order N with m components, m > floor(N/2) must contain an isolated vertex since it is impossible to partition N vertices in floor(N/2) + 1 or more trees without giving only one vertex to a tree.

Examples

			a(12) = 1 because 5 nodes can be partitioned into two trees only in one way: one tree gets 3 nodes and the other tree gets 2. Since A000055(3) = A000055(2) = 1, there is only one forest. (The forests of order less than or equal to 5 are depicted in the Weisstein link.)

Links

Crossrefs

Cf. A033185, A095133, A105786, A105821.

Formula

a(n) = sum over the partitions of N: 1K1 + 2K2 + ... + NKN, with exactly m parts and no part equal to 1, of Product_{i=1..N} binomial(A000055(i)+Ki-1, Ki).
G.f.: 1/Product_{i>=2}(1 - x*y^i)^A000055(i). - Vladeta Jovovic, Apr 27 2005
Showing 1-3 of 3 results.

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