Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #11 Oct 11 2020 04:47:07
%S 1,4,23,178,1763,21504,313585,5342068,104376201,2304582544,
%T 56807530871,1547599725720,46202052688603,1500629138909632,
%U 52697989385197137,1990117967149595824,80440669725095395025,3465573101368534916928
%N 3rd hyperbinomial transform of A001858; also the hyperbinomial transform of A089462.
%C A001858 enumerates forests of labeled trees with n nodes and shifts 1 place left under the hyperbinomial transform.
%H G. C. Greubel, <a href="/A089465/b089465.txt">Table of n, a(n) for n = 0..385</a>
%F a(n) = Sum_{k=0..n} 3*(n-k+3)^(n-k-1)*C(n, k)*A001858(k).
%F a(n) = Sum_{m=0..(n+1)} ( Sum_{j=0..m} C(m, j)*C(n, n-m-j+1)*(n+3)^(n-m-j+1)*(m+j)!/(-2)^j )/m!.
%F a(n) ~ 3 * exp(7/2) * n^(n-1). - _Vaclav Kotesovec_, Oct 11 2020
%t Table[Sum[Sum[Binomial[m, j]*Binomial[n, n - m - j + 1]*(n + 3)^(n - m - j + 1)*(m + j)!/(-2)^j, {j, 0, m}]/m!, {m, 0, n + 1}], {n, 0, 50}] (* _G. C. Greubel_, Nov 18 2017 *)
%o (PARI) a(n)=if(n<0,0,sum(m=0,n+1,sum(j=0,m,binomial(m,j)*binomial(n,n-m-j+1)*(n+3)^(n-m-j+1)*(m+j)!/(-2)^j)/m!))
%Y Cf. A001858, A089462, A089463.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 05 2003