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.

A244454 Number T(n,k) of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 7, 1, 0, 1, 0, 17, 2, 0, 0, 1, 0, 42, 4, 1, 0, 0, 1, 0, 105, 7, 2, 0, 0, 0, 1, 0, 267, 15, 2, 1, 0, 0, 0, 1, 0, 684, 28, 4, 2, 0, 0, 0, 0, 1, 0, 1775, 56, 7, 2, 1, 0, 0, 0, 0, 1, 0, 4639, 110, 12, 2, 2, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 28 2014

Keywords

Comments

T(1,0) = 1 by convention.
Sum_{i=2..n-1} T(n,i) = A001678(n+1) for n>1.

Examples

			The A000081(5) = 9 rooted trees with 5 nodes sorted by minimal outdegree of inner nodes are:
: 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   o o o o o     : o   o   :         :
: |  / \  |     |                       :         :         :
: o o   o o     o                       :         :         :
: |                                     :         :         :
: o                                     :         :         :
:                                       :         :         :
: ------------------1------------------ : ---2--- : ---4--- :
Thus row 5 = [0, 7, 1, 0, 1].
Triangle T(n,k) begins:
  1;
  0,    1;
  0,    1,   1;
  0,    3,   0,  1;
  0,    7,   1,  0, 1;
  0,   17,   2,  0, 0, 1;
  0,   42,   4,  1, 0, 0, 1;
  0,  105,   7,  2, 0, 0, 0, 1;
  0,  267,  15,  2, 1, 0, 0, 0, 1;
  0,  684,  28,  4, 2, 0, 0, 0, 0, 1;
  0, 1775,  56,  7, 2, 1, 0, 0, 0, 0, 1;
  0, 4639, 110, 12, 2, 2, 0, 0, 0, 0, 0, 1;

Links

Crossrefs

Columns k=0-10 give: A063524, A244455, A244456, A244457, A244458, A244459, A244460, A244461, A244462, A244463, A244464.
Row sums give A000081.
Cf. A001678, A244372, A244530 (ordered unlabeled rooted trees).

Programs

  • Maple
    b:= proc(n, i, t, k) option remember; `if`(n=0, `if`(t in [0, k],
      1, 0), `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
      b(n-i*j, i-1, max(0, t-j), k), j=0..n/i)))
    end:
T:= (n, k)-> b(n-1$2, k$2) -`if`(n=1 and k=0, 0, b(n-1$2, k+1$2)):
seq(seq(T(n, k), k=0..n-1), n=1..14);
  • Mathematica
    b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[i<1, 0, Sum[Binomial[b[i-1, i-1, k, k]+j-1, j]* b[n-i*j, i-1, Max[0, t-j], k], {j, 0, n/i}]]]; T[n_, k_] := b[n-1, n-1, k, k] - If[n == 1 && k == 0, 0, b[n-1, n-1, k+1, k+1]]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 08 2015, translated from Maple *)

A246403 Decimal expansion of a constant related to series-reduced trees.

Original entry on oeis.org

2, 1, 8, 9, 4, 6, 1, 9, 8, 5, 6, 6, 0, 8, 5, 0, 5, 6, 3, 8, 8, 7, 0, 2, 7, 5, 7, 7, 1, 1, 4, 5, 4, 4, 9, 6, 7, 3, 3, 1, 7, 0, 8, 7, 4, 4, 2, 3, 8, 4, 9, 0, 3, 0, 1, 0, 5, 0, 2, 7, 3, 4, 2, 5, 3, 5, 7, 1, 5, 6, 2, 5, 7, 0, 4, 2, 2, 1, 2, 2, 6, 3, 0, 0, 8, 5, 8, 6, 0, 7, 8, 4, 8, 1, 9, 3, 3, 3, 0, 8, 3, 2, 0, 3, 8
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 25 2014

Keywords

Examples

			2.189461985660850563887027577114544967331...

References

  • Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 302 and 561.

Crossrefs

Cf. A000014, A001678, A001679, A059123, A244456, A198518.

Formula

Equals lim n -> infinity A000014(n)^(1/n).
Equals lim n -> infinity A001678(n)^(1/n).
Equals lim n -> infinity A001679(n)^(1/n).
Equals lim n -> infinity A059123(n)^(1/n).
Equals lim n -> infinity A244456(n)^(1/n).
Equals lim n -> infinity A198518(n)^(1/n).

Extensions

More terms from Vaclav Kotesovec, Sep 03 2014 and Dec 26 2020

A244531 Number of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 2.

Original entry on oeis.org

1, 0, 2, 5, 11, 28, 78, 201, 532, 1441, 3895, 10569, 28926, 79493, 219226, 607189, 1687880, 4706737, 13165215, 36929595, 103860429, 292808814, 827392709, 2342964435, 6647953886, 18898472568, 53818654942, 153518738980, 438602656951, 1254943919799, 3595714927194
Offset: 3

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 29 2014

Keywords

Links

Crossrefs

Column k=2 of A244530.
Cf. A244456.

Programs

  • Maple
    b:= proc(n, t, k) option remember; `if`(n=0,
      `if`(t in [0, k], 1, 0), `if`(t>n, 0, add(b(j-1, k$2)*
       b(n-j, max(0, t-1), k), j=1..n)))
    end:
a:= n-> b(n-1, 2$2) -b(n-1, 3$2):
seq(a(n), n=3..50);
  • Mathematica
    b[n_, t_, k_] := b[n, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[t > n, 0, Sum[b[j - 1, k, k]*b[n - j, Max[0, t - 1], k], {j, 1, n}]]]; T[n_, k_] := b[n - 1, k, k] - If[n == 1 && k == 0, 0, b[n - 1, k + 1, k + 1]]; a[n_] := b[n - 1, 2, 2] - b[n - 1, 3, 3]; Table[a[n], {n, 3, 50}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

Formula

Recurrence: (n-2)*n*(n+1)*(31556*n^6 - 602602*n^5 + 4562565*n^4 - 17272550*n^3 + 33523297*n^2 - 29665770*n + 7578864)*a(n) = -2*(n-4)*n*(15778*n^6 - 93541*n^5 - 718683*n^4 + 7746097*n^3 - 25426183*n^2 + 35870760*n - 18623988)*a(n-1) + 2*(189336*n^9 - 4357178*n^8 + 42198478*n^7 - 222932639*n^6 + 692179375*n^5 - 1246825745*n^4 + 1121148607*n^3 - 95771898*n^2 - 622360656*n + 342066240)*a(n-2) + 4*(15778*n^9 - 301301*n^8 + 2556736*n^7 - 13524389*n^6 + 51959635*n^5 - 145042550*n^4 + 255185823*n^3 - 199177680*n^2 - 62590212*n + 146335680)*a(n-3) - 2*(n-4)*(63112*n^8 - 1252538*n^7 + 9554713*n^6 - 31464554*n^5 + 11620330*n^4 + 221568106*n^3 - 627283143*n^2 + 624591414*n - 146644560)*a(n-4) - 4*(n-5)*(n-4)*(504896*n^7 - 9428629*n^6 + 69275668*n^5 - 250040744*n^4 + 437755491*n^3 - 253595994*n^2 - 179277570*n + 187109352)*a(n-5) - 69*(n-6)*(n-5)*(n-4)*(31556*n^6 - 413266*n^5 + 2022895*n^4 - 4417190*n^3 + 3528357*n^2 + 989760*n - 1844640)*a(n-6). - Vaclav Kotesovec, Jul 02 2014
a(n) ~ 3^(n+1/2) / (8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 02 2014
Showing 1-3 of 3 results.

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