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.

A215977 Number T(n,k) of simple unlabeled graphs on n nodes with exactly k connected components that are trees or cycles; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 3, 3, 1, 1, 0, 4, 5, 3, 1, 1, 0, 7, 10, 6, 3, 1, 1, 0, 12, 17, 12, 6, 3, 1, 1, 0, 24, 33, 23, 13, 6, 3, 1, 1, 0, 48, 62, 47, 25, 13, 6, 3, 1, 1, 0, 107, 127, 92, 53, 26, 13, 6, 3, 1, 1, 0, 236, 267, 189, 106, 55, 26, 13, 6, 3, 1, 1
Offset: 0

Views

Author

Alois P. Heinz, Aug 29 2012

Keywords

Examples

			T(4,1) = 3: .o-o.  .o-o.  .o-o.
            .| |.  .|  .  .|\ .
            .o-o.  .o-o.  .o o.
.
T(4,2) = 3: .o-o.  .o-o.  .o-o.
            .|/ .  .|  .  .   .
            .o o.  .o o.  .o-o.
.
T(5,1) = 4: .o-o-o.  .o-o-o.  .o-o-o.  .o-o-o.
            .|  / .  .|    .  .| |  .  . /|  .
            .o-o  .  .o-o  .  .o o  .  .o o  .
.
T(5,2) = 5: .o-o o.  .o-o o.  .o-o o.  .o o-o.  .o o-o.
            .| |  .  .|    .  .|\   .  .|\   .  .|    .
            .o-o  .  .o-o  .  .o o  .  .o-o  .  .o-o  .
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  1,  1;
  0,  3,  3,  1,  1;
  0,  4,  5,  3,  1,  1;
  0,  7, 10,  6,  3,  1,  1;
  0, 12, 17, 12,  6,  3,  1,  1;
  ...

Links

Crossrefs

Columns k=0-10 give: A000007, A215981, A215982, A215983, A215984, A215985, A215986, A215987, A215988, A215989, A215980.
Row sums give: A215978.
Limiting sequence of reversed rows gives: A215979.
The labeled version of this triangle is A215861.

Programs

  • Mathematica
    b[n_] := b[n] = If[n <= 1, n, Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}]/(n-1)];
g[n_] := g[n] = If[n>2, 1, 0]+b[n]-(Sum [b[k]*b[n-k], {k, 0, n}] - If[Mod[n, 2] == 0, b[n/2], 0])/2;
p[n_, i_, t_] := p[n, i, t] = If[nJean-François Alcover, Dec 04 2014, after Alois P. Heinz *)
  • Python
    from sympy.core.cache import cacheit
from sympy import binomial, divisors
@cacheit
def b(n): return n if n<2 else sum([sum([d*b(d) for d in divisors(j)])*b(n - j) for j in range(1, n)])//(n - 1)
@cacheit
def g(n): return (1 if n>2 else 0) + b(n) - (sum(b(k)*b(n - k) for k in range(n + 1)) - (b(n//2) if n%2==0 else 0))//2
@cacheit
def p(n, i, t): return 0 if nIndranil Ghosh, Aug 07 2017

A215978 Number of simple unlabeled graphs on n nodes with connected components that are trees or cycles.

Original entry on oeis.org

1, 1, 2, 4, 8, 14, 28, 52, 104, 206, 429, 903, 1982, 4430, 10206, 23966, 57522, 140236, 347302, 870682, 2207819, 5651437, 14590703, 37948338, 99358533, 261684141, 692906575, 1843601797, 4926919859, 13220064562, 35604359531, 96218568474, 260850911485
Offset: 0

Views

Author

Alois P. Heinz, Aug 29 2012

Keywords

Examples

			a(4) = 8:
.o-o.  .o-o.  .o-o.  .o-o.  .o-o.  .o-o.  .o-o.  .o o.
.| |.  .|  .  .|\ .  .|/ .  .|  .  .   .  .   .  .   .
.o-o.  .o-o.  .o o.  .o o.  .o o.  .o-o.  .o o.  .o o.

Links

Crossrefs

Row sum of A215977.
The labeled version is A144164. The inverse Euler transform is A215981.

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:
g:= proc(n) option remember; local k; `if`(n>2, 1, 0)+ b(n)-
      (add(b(k)*b(n-k), k=0..n) -`if`(irem(n, 2)=0, b(n/2), 0))/2
    end:
p:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i)+j-1,j)*p(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> p(n, n):
seq(a(n), n=0..40);
  • Mathematica
    b[n_] := b[n] = If[n <= 1, n, (Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n - j], {j, 1, n - 1}])/(n - 1)];
g[n_] := g[n] = If[n > 2, 1, 0] + b[n] - (Sum[b[k]*b[n - k], {k, 0, n}] - If[Mod[n, 2] == 0, b[n/2], 0])/2;
p[n_, i_] := p[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[g[i] + j - 1, j]*p[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := p[n, n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 21 2017, translated from Maple *)
  • Python
    from sympy.core.cache import cacheit
from sympy import binomial, divisors
@cacheit
def b(n): return n if n<2 else sum([sum([d*b(d) for d in divisors(j)])*b(n - j) for j in range(1, n)])//(n - 1)
@cacheit
def g(n): return (1 if n>2 else 0) + b(n) - (sum([b(k)*b(n - k) for k in range(n + 1)]) - (b(n//2) if n%2==0 else 0))//2
@cacheit
def p(n, i): return 1 if n==0 else 0 if i<1 else sum([binomial(g(i) + j - 1, j)*p(n - i*j, i - 1) for j in range(n//i + 1)])
def a(n): return p(n, n)
print([a(n) for n in range(41)]) # Indranil Ghosh, Aug 07 2017

Formula

a(n) = Sum_{k=0..n} A215977(n,k).
a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.95576528565199497471481752412... is Otter's rooted tree constant, and c = 1.085767435235426664262830616636... . - Vaclav Kotesovec, Mar 22 2017

A215851 Number of simple labeled graphs on n nodes with exactly 1 connected component that is a tree or a cycle.

Original entry on oeis.org

1, 1, 4, 19, 137, 1356, 17167, 264664, 4803129, 100181440, 2359762091, 61937322624, 1792399894837, 56697025885696, 1946238657504975, 72058247875111936, 2862433512904759793, 121439708940308299776, 5480390058971655049939, 262144060822550204416000
Offset: 1

Views

Author

Alois P. Heinz, Aug 25 2012

Keywords

Examples

			a(3) = 4:
.1-2.  .1-2.  .1-2.  .1 2.
.|/ .  .|  .  . / .  .|/ .
.3...  .3...  .3...  .3...

Links

Crossrefs

Column k=1 of A215861.
The unlabeled version is A215981.
Cf. A000272, A001710.

Programs

  • Maple
    a:= n-> `if`(n<3, 1, (n-1)!/2+n^(n-2)):
seq(a(n), n=1..25);

Formula

a(1) = a(2) = 1, a(n) = A000272(n) + A001710(n-1) = n^(n-2) + (n-1)!/2 for n>2.
Showing 1-3 of 3 results.

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