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.

A005753 Number of rooted identity matched trees with n nodes.

Original entry on oeis.org

1, 2, 5, 18, 66, 266, 1111, 4792, 21124, 94888, 432415, 1994828, 9296712, 43706722, 207030398, 987130456, 4733961435, 22819241034, 110500644857, 537295738556, 2622248720234, 12840953621208, 63074566121245, 310693364823376, 1534374047239554, 7595642577152762
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Also number of rooted identity trees with n nodes and 2-colored non-root nodes. - Christian G. Bower, Apr 15 1998

Examples

			G.f.: A(x) = x + 2*x^2 + 5*x^3 + 18*x^4 + 66*x^5 + 266*x^6 + ...
where A(x) = x*(1+x)^2*(1+x^2)^4*(1+x^3)^10*(1+x^4)^36*(1+x^5)^132*... (the exponents are A038077(n), n>=1).

References

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

Links

Crossrefs

Cf. A038077, A246312.
Column k=2 of A255517.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(2*a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n=1, 1, b((n-1)$2)):
seq(a(n), n=1..40);  # Alois P. Heinz, Aug 01 2013
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[2*a[i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := If[n == 1, 1, b[n-1, n-1]]; Table[a[n] // FullSimplify, {n, 1, 30}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)
  • PARI
    {a(n)=polcoeff(x*prod(k=1, n-1, (1+x^k+x*O(x^n))^(2*a(k))), n)} /* Paul D. Hanna */

Formula

G.f.: x*Product_{n>=1} (1 + x^n)^(2*a(n)) = Sum_{n>=1} a(n)*x^n. - Paul D. Hanna, Dec 31 2011
a(n) ~ c * d^n / n^(3/2), where d = A246312 = 5.249032491228170579164952216..., c = 0.192066288645200371237879149260484794708740197522264442948290580404909605849... - Vaclav Kotesovec, Aug 25 2014, updated Dec 26 2020
G.f. A(x) satisfies: A(x) = x*exp(2*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k). - Ilya Gutkovskiy, Apr 13 2019

A246312 Decimal expansion of a constant related to identity matched trees.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Aug 25 2014

Keywords

Examples

			5.249032491228170579164952216184309265343086337648736503202233186059585565...

Crossrefs

Cf. A005753, A005754, A005755, A102755, A038078, A038077.

Formula

Equals lim n -> infinity A005753(n)^(1/n).
Equals lim n -> infinity A005754(n)^(1/n).
Equals lim n -> infinity A005755(n)^(1/n).
Equals lim n -> infinity A102755(n)^(1/n).
Equals lim n -> infinity A038078(n)^(1/n).

Extensions

More terms from Vaclav Kotesovec, Sep 06 2014, Feb 24 2015 and Dec 26 2020

A038080 Number of identity trees with 3-colored nodes.

Original entry on oeis.org

1, 3, 3, 9, 39, 189, 981, 5490, 31674, 189954, 1170126, 7382745, 47494197, 310712808, 2061987642, 13855192866, 94113385437, 645424668666, 4464027720900, 31110200069511, 218292811705458, 1541172223659249
Offset: 0

Views

Author

Christian G. Bower, Jan 04 1999

Keywords

Links

Crossrefs

Cf. A000220, A038077-A038079, A052757, A255517.

Formula

G.f.: B(x) - B^2(x)/2 - B(x^2)/2, where B(x) is g.f. for A038079.
a(n) ~ c * d^n / n^(5/2), where d = 7.969494030514425004826375511986491746399264355846412073489715938... and c = 0.3712461766927875417276388215355520756010680416348018056669... - Vaclav Kotesovec, Dec 26 2020

A038078 Number of identity trees with 2-colored nodes.

Original entry on oeis.org

1, 2, 1, 2, 6, 20, 69, 270, 1026, 4120, 16794, 70230, 298306, 1288912, 5642559, 25007756, 111998920, 506348902, 2308338456, 10602357346, 49026021552, 228085486580, 1067020210339, 5016982766202, 23698640081356, 112422573858292, 535414026652828, 2559204304109868
Offset: 0

Views

Author

Christian G. Bower, Jan 04 1999

Keywords

Links

Crossrefs

Cf. A000220. A038077-A038080.
Cf. A246312.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(2*b(i-1$2), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n=0, 1, 2*b(n-1$2) -2*add(b(j-1$2)*b(n-j-1$2)
        , j=1..n-1) -`if`(irem(n, 2, 'r')=0, b(r-1$2), 0)):
seq(a(n), n=0..35);  # Alois P. Heinz, Aug 02 2013
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[2*b[i-1, i-1], j]*b[n-i*j, i-1], {j, 0, n/i}]]];
a[n_] := If[n==0, 1, 2*b[n-1, n-1] - 2*Sum[b[j-1, j-1]*b[n-j-1, n-j-1], {j, 1, n-1}] - If[Mod[n, 2]==0, r=n/2; b[r-1, r-1], 0]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Mar 01 2016, after Alois P. Heinz *)

Formula

G.f.: B(x) - B^2(x)/2 - B(x^2)/2, where B(x) is g.f. for A038077.
a(n) ~ c * d^n / n^(5/2), where d = A246312 = 5.2490324912281705791649522161843092..., c = 0.356142078281568492877259973613... . - Vaclav Kotesovec, Sep 06 2014

A038079 Number of rooted identity trees with 3-colored nodes.

Original entry on oeis.org

3, 9, 36, 192, 1089, 6642, 42129, 275571, 1844028, 12567762, 86924745, 608612814, 4305284400, 30723593751, 220917003516, 1599013642359, 11641144003245, 85186969539435, 626245146267393, 4622806865390823, 34251800667936939, 254640579219212457
Offset: 1

Views

Author

Christian G. Bower, Jan 04 1999

Keywords

Links

Crossrefs

Cf. A004111, A038077-A038080.
Cf. A052757.

Formula

Shifts left and divides by 3 under Weigh transform.
a(n) = 3 * A052757(n).
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.