A317677 Fixed point of a shifted hypertree transform.

1, 1, 4, 32, 402, 7038, 160114, 4522578, 153640590, 6132546770, 282517271694, 14812447505646, 873934551644074, 57486823088667270, 4183353479821220130, 334572221351085006242, 29242220614539638127294, 2779426070382982579163202, 286058737295150226682469518

Offset: 1

Author

Gus Wiseman, Aug 04 2018

Keywords

• nonn
• eigen

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

• Alois P. Heinz, Table of n, a(n) for n = 1..305

Crossrefs

Cf. A000272, A030019, A048143, A134954, A275307, A293510, A317631, A317632, A317634, A317635, A317671.

Programs

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 *)