A058311 Number of nodes at n-th level in tree in which top node is 1; each node k has children labeled k, k+1, ..., (k+1)^2 at next level.

1, 4, 48, 7918, 463339346, 7134188685100826388, 13246386641449904934758023373599438217628, 643152870463337226096320122089499144560533929707886143570111588898313745804013188842

Offset

0,2

Comments

Triggered by a comment from Michael Kleber, Dec 08 2009, who said: The algorithm in my paper with Cook lets you compute the equivalent sequence where the children of a node labeled (k) are labeled with all the integers in the interval [p(k), q(k)] where p,q are any polynomials you like (in the paper, p(k)=k+1 and q(k)=2k). For a bunch of p,q the resulting sequence is well known, e.g., p(k)=1, q(k)=k+1 is the Catalan numbers.

Links

Table of n, a(n) for n=0..7.

M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.

Maple

M:=4;
L[0]:=[1]; a[0]:=1;
for n from 1 to M do
L[n]:=[];
t1:=L[n-1];
tc:=nops(t1);
for i from 1 to tc do
t2:=t1[i];
for j from t2 to (t2+1)^2 do
L[n]:=[op(L[n]), j]; od:
a[n]:=nops(L[n]);
#lprint(n, L[n], a[n]);
od:
od:
[seq(a[n], n=0..M)];
# See the reference for a better way to compute this!
p := proc(n, k) option remember; local j ; if n = 1 then k^2+k+2; # (k+1)^2-(k-1) else sum( procname(n-1, j), j=k..(k+1)^2) ; fi; expand(%) ; end proc:
A058311 := proc(n) if n = 0 then 1 ; else subs(k=1, p(n, k)) ; fi; end proc:
for n from 0 do printf("%d, \n", A058311(n)) ; od: # R. J. Mathar, May 04 2009

Mathematica

p[n_, k_] := p[n, k] = If[n == 1, k^2+k+2, Sum[p[n-1, j], {j, k, (k+1)^2}]];
a[n_] := If[n == 0, 1, p[n, 1]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 7}] (* Jean-François Alcover, Jun 26 2023, after R. J. Mathar *)

Crossrefs

Cf. A008934, A058222, A147780, A147794.

Sequence in context: A123373 A264265 A132510 * A189347 A248558 A372052

Adjacent sequences: A058308 A058309 A058310 * A058312 A058313 A058314

Keyword

nonn

Author

N. J. A. Sloane, Dec 09 2000

Extensions

Corrected, with Maple program, by N. J. A. Sloane, May 03 2009. Thanks to Max Alekseyev for pointing out that something was wrong.

Replaced a(4), added three more terms - R. J. Mathar, May 04 2009

Status

approved