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 1-5 of 5 results.

A122086 Number of unlabeled free bicolored trees with n nodes (the colors are not interchangeable).

Original entry on oeis.org

2, 1, 2, 3, 6, 10, 22, 42, 94, 203, 470, 1082, 2602, 6270, 15482, 38525, 97258, 247448, 635910, 1645411, 4289010, 11245670, 29656148, 78595028, 209273780, 559574414, 1502130920, 4046853091, 10939133170, 29661655793
Offset: 1

Views

Author

N. J. A. Sloane, Oct 19 2006

Keywords

References

  • R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1978.

Links

Crossrefs

Row sums of A122085.
Antidiagonal sums of A329054.
Same as A125702 except for n = 1.
Cf. A000055, A000081.

Programs

  • PARI
    \\ here TreeGf is A000081 as g.f.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={Vec(2*TreeGf(n) - TreeGf(n)^2)} \\ Andrew Howroyd, Nov 02 2019

Formula

For n even, a(n) = 2*A000055(n) - A000081(n/2), for n odd, a(n) = 2*A000055(n).
G.f.: 2*f(x) - f(x)^2 where f(x) is the g.f. of A000081. - Andrew Howroyd, Nov 02 2019

Extensions

Edited by Christian G. Bower and Franklin T. Adams-Watters, Jan 05 2007

A119857 Number of equicolored (unrooted) trees on 2n nodes.

Original entry on oeis.org

1, 1, 4, 14, 65, 316, 1742, 10079, 61680, 391473, 2565262, 17237962, 118341446, 827194809, 5872518213, 42256545977, 307681822711, 2263881127801, 16813356777456, 125917441081662, 950148951332802, 7218810159035143, 55187741462110393, 424318236236124092
Offset: 1

Views

Author

N. J. A. Sloane, Aug 04 2006

Keywords

Comments

For precise definition, recurrence and asymptotics see the Pippenger reference.

Links

Crossrefs

Main diagonal of A329054.
Cf. A119855, A119856.

Programs

  • PARI
    \\ R is b.g.f of rooted trees x nodes, y in one part
R(n)={my(A=O(x)); for(j=1, 2*n, A = if(j%2,1,y)*x*exp(sum(i=1, j, 1/i * subst(subst(A + x * O(x^(j\i)), x, x^i), y, y^i)))); A};
seq(n)={my(A=Pol(R(n))); my(r(x,y)=substvec(A, ['x,'y], [x,y/x])); Vec(polcoeff((r(x,y/x) + r(y/x,x) - r(x,y/x)*r(y/x,x)), 0) + O(y*y^n))} \\ Andrew Howroyd, May 23 2018

Extensions

Terms a(8) and beyond from Andrew Howroyd, May 21 2018

A122085 Triangle read by rows: T(n,k) = number of unlabeled free bicolored trees with n nodes (n >= 1) and k (1 <= k <= n-1, except k=0 or 1 if n=1, k=1 if n=2) nodes of one color and n-k nodes of the other color (the colors are not interchangeable).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 7, 7, 3, 1, 1, 3, 10, 14, 10, 3, 1, 1, 4, 14, 28, 28, 14, 4, 1, 1, 4, 19, 45, 65, 45, 19, 4, 1, 1, 5, 24, 73, 132, 132, 73, 24, 5, 1, 1, 5, 30, 105, 242, 316, 242, 105, 30, 5, 1, 1, 6, 37, 152, 412, 693, 693, 412, 152
Offset: 1

Views

Author

N. J. A. Sloane, Oct 19 2006

Keywords

Examples

			K M N gives the number N of unlabeled free bicolored trees with K nodes of one color and M nodes of the other color.
0 1 1
1 0 1
Total( 1) = 2
1 1 1
Total( 2) = 1
1 2 1
2 1 1
Total( 3) = 2
1 3 1
2 2 1
3 1 1
Total( 4) = 3
1 4 1
2 3 2
3 2 2
4 1 1
Total( 5) = 6
1 5 1
2 4 2
3 3 4
4 2 2
5 1 1
Total( 6) = 10
.
From _Andrew Howroyd_, Nov 02 2019: (Start)
Triangle for n >= 2, 1 <= k < n:
   2 | 1;
   3 | 1, 1;
   4 | 1, 1,  1;
   5 | 1, 2,  2,   1;
   6 | 1, 2,  4,   2,   1;
   7 | 1, 3,  7,   7,   3,   1;
   8 | 1, 3, 10,  14,  10,   3,   1;
   9 | 1, 4, 14,  28,  28,  14,   4,   1;
  10 | 1, 4, 19,  45,  65,  45,  19,   4,  1;
  11 | 1, 5, 24,  73, 132, 132,  73,  24,  5, 1;
  12 | 1, 5, 30, 105, 242, 316, 242, 105, 30, 5, 1;
  ...
(End)

References

  • R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1978.

Links

Crossrefs

Row sums give A122086.
Cf. A329054 (regular array with same data).

A329052 Array read by antidiagonals: T(n,m) is the number of unlabeled bicolored acyclic graphs with n nodes of one color and m of the other.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 21, 15, 6, 1, 1, 7, 21, 38, 38, 21, 7, 1, 1, 8, 28, 62, 82, 62, 28, 8, 1, 1, 9, 36, 95, 158, 158, 95, 36, 9, 1, 1, 10, 45, 138, 278, 356, 278, 138, 45, 10, 1, 1, 11, 55, 192, 459, 724, 724, 459, 192, 55, 11, 1
Offset: 0

Views

Author

Andrew Howroyd, Nov 02 2019

Keywords

Comments

The two color classes are not interchangeable. Adjacent nodes cannot have the same color.

Examples

			Array begins:
=======================================================
n\m | 0  1   2    3    4     5     6      7      8
----+--------------------------------------------------
  0 | 1, 1,  1,   1,   1,    1,    1,     1,     1, ...
  1 | 1, 2,  3,   4,   5,    6,    7,     8,     9, ...
  2 | 1, 3,  6,  10,  15,   21,   28,    36,    45, ...
  3 | 1, 4, 10,  21,  38,   62,   95,   138,   192, ...
  4 | 1, 5, 15,  38,  82,  158,  278,   459,   716, ...
  5 | 1, 6, 21,  62, 158,  356,  724,  1359,  2388, ...
  6 | 1, 7, 28,  95, 278,  724, 1690,  3612,  7143, ...
  7 | 1, 8, 36, 138, 459, 1359, 3612,  8731, 19404, ...
  8 | 1, 9, 45, 192, 716, 2388, 7143, 19404, 48213, ...
  ...

Links

Crossrefs

Main diagonal is A329055.
Antidiagonal sums are A329053.
The equivalent array for labeled nodes is A328887.
Cf. A329054.

Programs

  • PARI
    EulerXY(A)={my(j=serprec(A,x)); exp(sum(i=1, j, 1/i * subst(subst(A + x * O(x^(j\i)), x, x^i), y, y^i)))}
R(n)={my(A=O(x)); for(j=1, 2*n, A = if(j%2, 1, y)*x*EulerXY(A)); A};
P(n)={my(r1=R(n), r2=x*EulerXY(r1), s=r1+r2-r1*r2); Vec(EulerXY(s))}
{ my(A=P(10)); for(n=0, #A\2, for(k=0, #A\2, print1(polcoef(A[n+k+1], k), ", ")); print) }

A122087 Triangle read by rows: T(n,k) = number of unlabeled free bicolored trees with n nodes (n >= 1) and k (1 <= k <= floor(n/2), except k = 0 if n = 1 ) nodes of one color and n-k nodes of the other color (the colors are interchangeable).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 3, 1, 3, 7, 1, 3, 10, 9, 1, 4, 14, 28, 1, 4, 19, 45, 37, 1, 5, 24, 73, 132, 1, 5, 30, 105, 242, 168, 1, 6, 37, 152, 412, 693, 1, 6, 44, 204, 660, 1349, 895, 1, 7, 52, 274, 1008, 2472, 3927, 1, 7, 61, 351, 1479, 4219, 8105, 5097, 1, 8
Offset: 1

Views

Author

N. J. A. Sloane, Oct 19 2006

Keywords

Examples

			K M N gives the number N of unlabeled free bicolored trees with K nodes of one color and M nodes of the other color.
0 1 1
Total( 1) = 1
1 1 1
Total( 2) = 1
1 2 1
Total( 3) = 1
1 3 1
2 2 1
Total( 4) = 2
1 4 1
2 3 2
Total( 5) = 3
1 5 1
2 4 2
3 3 3
Total( 6) = 6
1 6 1
2 5 3
3 4 7
Total( 7) = 11
1 7 1
2 6 3
3 5 10
4 4 9
Total( 8) = 23
From _Andrew Howroyd_, Apr 05 2023: (Start)
Triangle begins:
  n\k| 0 1  2   3    4    5    6
 ----+----------------------------
   1 | 1;
   2 | . 1;
   3 | . 1;
   4 | . 1, 1;
   5 | . 1, 2;
   6 | . 1, 2,  3;
   7 | . 1, 3,  7;
   8 | . 1, 3, 10,   9;
   9 | . 1, 4, 14,  28;
  10 | . 1, 4, 19,  45,  37;
  11 | . 1, 5, 24,  73, 132;
  12 | . 1, 5, 30, 105, 242, 168;
    ...
(End)

References

  • R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1978.

Links

Crossrefs

Row sums give A000055.
Cf. A119856, A329054, A362019 (labeled version).

Formula

T(n,k) = A329054(k, n-k) for 2*k < n; T(2*n,n) = A119856(n). - Andrew Howroyd, Apr 04 2023
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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