%I #8 Feb 26 2017 10:54:31
%S 1,2,14,178,3270,78316,2308876,80775780,3269037596,150194207800,
%T 7721544428136,439128840082648,27369393580944520,1855079496872679312,
%U 135846807056384160080,10688153505317713069936,899138432350085506208784,80536073356838110790279200
%N a(n) = n-th number of the n-th iteration of the hyperbinomial transform on sequence A001858 (number of forests of trees on n labeled nodes).
%H Alois P. Heinz, <a href="/A252727/b252727.txt">Table of n, a(n) for n = 0..200</a>
%p hymtr:= proc(p) proc(n, m) `if`(m=0, p(n), m*
%p add(p(k)*binomial(n, k) *(n-k+m)^(n-k-1), k=0..n))
%p end end:
%p f:= proc(n) option remember; add(add(binomial(n-1, n-m-j)*
%p binomial(m, j)*n^(n-m-j)*(m+j)!/(-2)^j/m!, j=0..m), m=0..n)
%p end:
%p A:= hymtr(f): a:= n-> A(n$2):
%p seq(a(n), n=0..20);
%t hymtr[p_] := Function[{n, m}, If[m==0, p[n], m*Sum[p[k]*Binomial[n, k]*(n - k + m)^(n-k-1), {k, 0, n}]]]; f[0] = 1; f[n_] := f[n] = Sum[ Sum[ Binomial[m, j] * Binomial[n-1, n-m-j]*n^(n-m-j)*(m+j)!/(-2)^j/m!, {j, 0, m}], {m, 0, n}]; A[0, _] = 1; A[1, k_] := k+1; A[n_, m_] := hymtr[f][n, n]; a[n_] := A[n, n]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Feb 26 2017, after _Alois P. Heinz_ *)
%Y Main diagonal of A144304.
%Y Cf. A001858.
%K nonn
%O 0,2
%A _Alois P. Heinz_, Dec 20 2014