login
A317677
Fixed point of a shifted hypertree transform.
5
1, 1, 4, 32, 402, 7038, 160114, 4522578, 153640590, 6132546770, 282517271694, 14812447505646, 873934551644074, 57486823088667270, 4183353479821220130, 334572221351085006242, 29242220614539638127294, 2779426070382982579163202, 286058737295150226682469518
OFFSET
1,3
COMMENTS
The hypertree transform H(a) of a sequence a is given by H(a)(n) = Sum_p n^(k-1) Prod_i a(|p_i|+1), where the sum is over all set partitions U(p_1, ..., p_k) = {1, ..., n-1}.
LINKS
MAPLE
b:= proc(n, k) option remember; `if`(n=0, 1/k, add(
a(j)*b(n-j, k)*binomial(n-1, j-1)*k, j=1..n))
end:
a:= n-> b(n-1, n):
seq(a(n), n=1..20); # Alois P. Heinz, Aug 21 2019
MATHEMATICA
numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
a[n_]:=a[n]=Sum[n^(Length[ptn]-1)*numSetPtnsOfType[ptn]*Product[a[s], {s, ptn}], {ptn, IntegerPartitions[n-1]}];
Array[a, 20]
(* Second program: *)
b[n_, k_] := b[n, k] = If[n == 0, 1/k, Sum[
a[j]*b[n - j, k]*Binomial[n - 1, j - 1]*k, {j, 1, n}]];
a[n_] := b[n - 1, n];
Array[a, 20] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)
KEYWORD
nonn,eigen
AUTHOR
Gus Wiseman, Aug 04 2018
STATUS
approved