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A317677 Fixed point of a shifted hypertree transform.

%I A317677 #12 May 10 2021 07:40:46
%S A317677 1,1,4,32,402,7038,160114,4522578,153640590,6132546770,282517271694,
%T A317677 14812447505646,873934551644074,57486823088667270,4183353479821220130,
%U A317677 334572221351085006242,29242220614539638127294,2779426070382982579163202,286058737295150226682469518
%N A317677 Fixed point of a shifted hypertree transform.
%C A317677 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}.
%H A317677 Alois P. Heinz, <a href="/A317677/b317677.txt">Table of n, a(n) for n = 1..305</a>
%p A317677 b:= proc(n, k) option remember; `if`(n=0, 1/k, add(
%p A317677       a(j)*b(n-j, k)*binomial(n-1, j-1)*k, j=1..n))
%p A317677     end:
%p A317677 a:= n-> b(n-1, n):
%p A317677 seq(a(n), n=1..20);  # _Alois P. Heinz_, Aug 21 2019
%t A317677 numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
%t A317677 a[n_]:=a[n]=Sum[n^(Length[ptn]-1)*numSetPtnsOfType[ptn]*Product[a[s],{s,ptn}],{ptn,IntegerPartitions[n-1]}];
%t A317677 Array[a,20]
%t A317677 (* Second program: *)
%t A317677 b[n_, k_] := b[n, k] = If[n == 0, 1/k, Sum[
%t A317677      a[j]*b[n - j, k]*Binomial[n - 1, j - 1]*k, {j, 1, n}]];
%t A317677 a[n_] := b[n - 1, n];
%t A317677 Array[a, 20] (* _Jean-François Alcover_, May 10 2021, after _Alois P. Heinz_ *)
%Y A317677 Cf. A000272, A030019, A048143, A134954, A275307, A293510, A317631, A317632, A317634, A317635, A317671.
%K A317677 nonn,eigen
%O A317677 1,3
%A A317677 _Gus Wiseman_, Aug 04 2018