The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A059123 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

A059123

Number of homeomorphically irreducible rooted trees (also known as series-reduced rooted trees, or rooted trees without nodes of degree 2) with n >= 1 nodes.
(history; published version)

#37 by N. J. A. Sloane at Mon Apr 04 10:15:15 EDT 2022

STATUS

proposed

approved

#36 by Eric W. Weisstein at Mon Apr 04 07:43:41 EDT 2022

STATUS

editing

proposed

#35 by Eric W. Weisstein at Fri Mar 25 14:17:46 EDT 2022

COMMENTS

Essentially the same as A001679. - Eric W. Weisstein, Mar 25 2022

STATUS

approved

editing

Discussion

Fri Apr 01

18:37

OEIS Server: This sequence has not been edited or commented on for a week
yet is not proposed for review.  If it is ready for review, please
visit https://oeis.org/draft/A059123 and click the button that reads
"These changes are ready for review by an OEIS Editor."

Thanks.
  - The OEIS Server

#34 by Vaclav Kotesovec at Sat Dec 26 11:19:28 EST 2020

STATUS

editing

approved

#33 by Vaclav Kotesovec at Sat Dec 26 11:19:21 EST 2020

FORMULA

a(n) ~ c * d^n / n^(3/2), where d = A246403 = 2.18946198566085056388702757711... and c = 0.4213018528699249210965028.421301852869924921096502830935802411658488216342994235732491571594804013.. . - Vaclav Kotesovec, Jun 26 2014

STATUS

approved

editing

#32 by Alois P. Heinz at Fri Jan 12 11:27:04 EST 2018

STATUS

proposed

approved

#31 by Jean-François Alcover at Fri Jan 12 11:26:14 EST 2018

STATUS

editing

proposed

#30 by Jean-François Alcover at Fri Jan 12 11:25:52 EST 2018

MATHEMATICA

max = 36; G001678[x_] := Sum[a[k]*x^k, {k, 0, max}]; a[0] = a[1] = a[3] = 0; a[2] = a[4] = a[5] = 1; coes = CoefficientList[ Series[G001678[x] - x^2/((1 + x)*Product[(1 - x^k)^a[k + 1], {k, 1, max}]), {x, 0, max}], x]; sol = Solve[Thread[coes == 0]][[1]]; f[x_] = G001678[x] /. sol; se = Series[ (1 + ((1 + x)/x)*f[x] - (f[x]^2 + f[x^2])/(2*x)), {x, 0, max}]; A059123 = Join[{0}, CoefficientList[se, x][[2 ;; -2]]] (* Jean-François Alcover, May 25 2012, from g.f. *)

terms = 36; (* F = G001678 *) F[_] = 0; Do[F[x_] = (x^2/(1 + x))*Exp[Sum[ F[x^k]/(k*x^k), {k, 1, j}]] + O[x]^j // Normal, {j, 1, terms + 1}];

G[x_] = 1 + ((1 + x)/x)*F[x] - (F[x]^2 + F[x^2])/(2*x) + O[x]^terms;

CoefficientList[G[x] - 1, x] (* Jean-François Alcover, May 25 2012, updated Jan 12 2018 *)

STATUS

approved

editing

Discussion

Fri Jan 12

11:26

Jean-François Alcover: Simplified Mma program.

#29 by Jon E. Schoenfield at Thu Mar 05 12:22:17 EST 2015

STATUS

editing

approved

#28 by Jon E. Schoenfield at Thu Mar 05 12:22:16 EST 2015

AUTHOR

Wolfdieter Lang , Jan 09 2001