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 1-6 of 6 results.

A055290 Triangle of trees with n nodes and k leaves, 2 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 1, 3, 4, 2, 1, 0, 1, 4, 8, 6, 3, 1, 0, 1, 5, 14, 14, 9, 3, 1, 0, 1, 7, 23, 32, 26, 12, 4, 1, 0, 1, 8, 36, 64, 66, 39, 16, 4, 1, 0, 1, 10, 54, 123, 158, 119, 60, 20, 5, 1, 0, 1, 12, 78, 219, 350, 325, 202, 83, 25, 5, 1, 0
Offset: 2

Views

Author

Christian G. Bower, May 09 2000

Keywords

Examples

			Triangle begins:
  n=2:  1
  n=3:  1   0
  n=4:  1   1   0
  n=5:  1   1   1   0
  n=6:  1   2   2   1   0
  n=7:  1   3   4   2   1   0
  n=8:  1   4   8   6   3   1   0
  n=9:  1   5  14  14   9   3   1   0
  n=10: 1   7  23  32  26  12   4   1   0
  n=11: 1   8  36  64  66  39  16   4   1   0
  n=12: 1  10  54 123 158 119  60  20   5   1   0
  n=13: 1  12  78 219 350 325 202  83  25   5   1   0

References

  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 80, Problem 3.9.

Links

Crossrefs

Row sums give A000055, row sums with weight k give A003228.
Columns 3 through 12: A001399(n-4), A055291, A055292, A055293, A055294, A055295, A055296, A055297, A055298, A055299.
Cf. A055300, A055301, A238416, A304222.
The labeled version is A055314.
Central column is A358107.
Left of central column is A359398.
Cf. A000088, A000272, A001349, A001433, A002494, A185650.

Programs

  • PARI
    EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
T(n)={my(u=[y]); for(n=2, n, u=concat([y], EulerMT(u))); my(r=x*Ser(u), v=Vec(r*(1-x+x*y) + (substvec(r,[x,y],[x^2,y^2]) - r^2)/2)); vector(n-1, k, Vecrev(v[1+k]/y^2, k))}
{ my(A=T(10)); for(n=1, #A, print(A[n])) }

Formula

G.f.: A(x, y)=(1-x+x*y)*B(x, y)+(1/2)*(B(x^2, y^2)-B(x, y)^2), where B(x, y) is g.f. of A055277.

A055897 a(n) = n*(n-1)^(n-1).

Original entry on oeis.org

1, 2, 12, 108, 1280, 18750, 326592, 6588344, 150994944, 3874204890, 110000000000, 3423740047332, 115909305827328, 4240251492291542, 166680102383370240, 7006302246093750000, 313594649253062377472, 14890324713954061755186, 747581753430634213933056
Offset: 1

Views

Author

Christian G. Bower, Jun 12 2000

Keywords

Comments

Total number of leaves in all labeled rooted trees with n nodes.
Number of endofunctions of [n] such that no element of [n-1] is fixed. E.g., a(3)=12: 123 -> 331, 332, 333, 311, 312, 313, 231, 232, 233, 211, 212, 213.
Number of functions f: {1, 2, ..., n} --> {1, 2, ..., n} such that f(1) != f(2), f(2) != f(3), ..., f(n-1) != f(n). - Warut Roonguthai, May 06 2006
Determinant of the n X n matrix ((2n, n^2, 0, ..., 0), (1, 2n, n^2, 0, ..., 0), (0, 1, 2n, n^2, 0, ..., 0), ..., (0, ..., 0, 1, 2n)). - Michel Lagneau, May 04 2010
For n > 1: a(n) = A240993(n-1) / A240993(n-2). - Reinhard Zumkeller, Aug 31 2014
Total number of points m such that f^(-1)(m) = {m}, (i.e., the preimage of m is the singleton set {m}) summed over all functions f:[n]->[n]. - Geoffrey Critzer, Jan 20 2022

Links

Crossrefs

Cf. A003227, A003228, A055314, A055540, A055541, A060226, A118537.
Cf. A240993. Diagonal of A265583.

Programs

Formula

E.g.f.: x/(1-T), where T=T(x) is Euler's tree function (see A000169).
a(n) = Sum_{k=1..n} A055302(n, k)*k.
a(n) = the n-th term of the (n-1)-th binomial transform of {1, 1, 4, 18, 96, ..., (n-1)*(n-1)!, ...} (cf. A001563). - Paul D. Hanna, Nov 17 2003
a(n) = (n-1)^(n-1) + Sum_{i=2..n} (n-1)^(n-i)*binomial(n-1, i-1)*(i-1) *(i-1)!. - Paul D. Hanna, Nov 17 2003
a(n) = [x^(n-1)] 1/(1 - (n-1)*x)^2. - Paul D. Hanna, Dec 27 2012
a(n) ~ exp(-1) * n^n. - Vaclav Kotesovec, Nov 14 2014

Extensions

Additional comments from Vladeta Jovovic, Mar 31 2001 and Len Smiley, Dec 11 2001

A055541 Total number of leaves (nodes of vertex degree 1) in all labeled trees with n nodes.

Original entry on oeis.org

0, 2, 6, 36, 320, 3750, 54432, 941192, 18874368, 430467210, 11000000000, 311249095212, 9659108818944, 326173191714734, 11905721598812160, 467086816406250000, 19599665578316398592, 875901453762003632658, 41532319635035234107392, 2082547005958224830656820
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

Equivalently, a(n) is the number of rooted labeled trees such that the root node has degree 1. - Geoffrey Critzer, Feb 07 2012

Links

Crossrefs

Cf. A003227, A003228, A055314, A055540, A055897.
Essentially the same as A061302.

Programs

  • Magma
    [0,2] cat [n*(n-1)^(n-2): n in [3..10]]; // G. C. Greubel, Nov 11 2017
  • Mathematica
    Join[{0,2}, Table[Sum[n!/k! StirlingS2[n-2,n-k] k, {k,2,n-1}], {n,3,20}]] (* Geoffrey Critzer, Nov 22 2011 *)
Join[{0,2}, Table[n*(n-1)^(n-2), {n,3,50}]] (* or *) Rest[With[{nmax = 40}, CoefficientList[Series[-x*LambertW[-x], {x, 0, nmax}], x]*Range[0, nmax]!]] (* G. C. Greubel, Nov 11 2017 *)
  • PARI
    for(n=1, 30, print1(if(n==1, 0, if(n==2, 2, n*(n-1)^(n-2))), ", ")) \\ G. C. Greubel, Nov 11 2017

Formula

From Vladeta Jovovic, Mar 31 2001: (Start)
a(n) = n*(n-1)^(n-2), n > 1.
E.g.f.: -x*LambertW(-x). (End)
a(n) = Sum_{k=1..n} (A055314(n, k)*k). - Christian G. Bower, Jun 12 2000
E.g.f.: x*T(x) where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, Feb 07 2012

Extensions

More terms from Christian G. Bower, Jun 12 2000

A003227 Endpoints (leaves) in rooted trees with n nodes.

Original entry on oeis.org

1, 1, 3, 8, 22, 58, 160, 434, 1204, 3341, 9363, 26308, 74376, 210823, 599832, 1710803, 4891876, 14015505, 40231632, 115669419, 333052242, 960219982, 2771707332, 8009222307, 23166563032, 67069289457, 194332834601
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of unlabeled rooted trees with n nodes and a distinguished leaf. - Gus Wiseman, Jul 31 2018

Examples

			The a(4) = 8 rooted trees with a distinguished leaf are (((O))), ((Oo)), ((oO)), (O(o)), (o(O)), (Ooo), (oOo), (ooO). - _Gus Wiseman_, Jul 31 2018

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000081, A003228, A004111, A038046, A055277, A317580.

Programs

  • Mathematica
    urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
Table[Sum[Length[Flatten[{t/.{}->1}]],{t,urt[n]}],{n,15}] (* Gus Wiseman, Jul 31 2018 *)

Formula

a(n) = Sum_{k=1..n} k*A055277(n, k).

Extensions

Corrected and extended with formula by Christian G. Bower, May 25 2000

A055540 Total number of leaves (nodes of vertex degree 1) in all graphs of n nodes.

Original entry on oeis.org

0, 2, 4, 14, 38, 153, 766, 6259, 88064, 2324157, 116563882, 11060411527, 1968703079886, 654492092481733, 406111248305672980, 471005105043787823717, 1023566652048387537072658, 4179937690541808658135640875, 32172436158252943170541450460638
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Links

Crossrefs

Cf. A003227, A003228, A055541, A327371.

Programs

Formula

a(n) = Sum_{k=1..n} k*A327371(n, k). - Andrew Howroyd, Sep 04 2019

Extensions

a(8) and a(9) from Eric W. Weisstein, Jun 02 2004
a(10) from Andrew Howroyd, Sep 04 2019
Terms a(11) and beyond from Andrew Howroyd, Jan 22 2021

A345133 Decimal expansion of the limit, as n approaches infinity, of the probability that a node is a leaf in a free tree with n nodes.

Original entry on oeis.org

4, 3, 8, 1, 5, 6, 2, 3, 5, 6, 6, 4
Offset: 0

Views

Author

Washington Bomfim, Aug 17 2021

Keywords

Examples

			0.438156235664...

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.6.3, p. 304.

Links

Crossrefs

Cf. A003228, A055543.

Formula

Equals lim_{n->oo} (A003228(n) / A055543(n)).
Showing 1-6 of 6 results.

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

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