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%I #24 Jun 26 2023 08:47:38
%S 1,4,48,7918,463339346,7134188685100826388,
%T 13246386641449904934758023373599438217628,
%U 643152870463337226096320122089499144560533929707886143570111588898313745804013188842
%N 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.
%C 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.
%H M. Cook and M. Kleber, <a href="https://doi.org/10.37236/1522">Tournament sequences and Meeussen sequences</a>, Electronic J. Comb. 7 (2000), #R44.
%p M:=4;
%p L[0]:=[1]; a[0]:=1;
%p for n from 1 to M do
%p L[n]:=[];
%p t1:=L[n-1];
%p tc:=nops(t1);
%p for i from 1 to tc do
%p t2:=t1[i];
%p for j from t2 to (t2+1)^2 do
%p L[n]:=[op(L[n]),j]; od:
%p a[n]:=nops(L[n]);
%p #lprint(n,L[n],a[n]);
%p od:
%p od:
%p [seq(a[n],n=0..M)];
%p # See the reference for a better way to compute this!
%p 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:
%p A058311 := proc(n) if n = 0 then 1 ; else subs(k=1, p(n,k)) ; fi; end proc:
%p for n from 0 do printf("%d,\n", A058311(n)) ; od: # _R. J. Mathar_, May 04 2009
%t p[n_, k_] := p[n, k] = If[n == 1, k^2+k+2, Sum[p[n-1, j], {j, k, (k+1)^2}]];
%t a[n_] := If[n == 0, 1, p[n, 1]];
%t Table[Print[n, " ", a[n]]; a[n], {n, 0, 7}] (* _Jean-François Alcover_, Jun 26 2023, after _R. J. Mathar_ *)
%Y Cf. A008934, A058222, A147780, A147794.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Dec 09 2000
%E Corrected, with Maple program, by _N. J. A. Sloane_, May 03 2009. Thanks to _Max Alekseyev_ for pointing out that something was wrong.
%E Replaced a(4), added three more terms - _R. J. Mathar_, May 04 2009
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