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A089462 2nd hyperbinomial transform of A001858. 4

%I

%S 1,3,14,93,822,9193,125292,2022555,37829468,805712859,19270873704,

%T 511742870653,14946235170120,476314240239633,16451368229689808,

%U 612254102183085627,24428043107239133712,1040281158638494489075

%N 2nd hyperbinomial transform of A001858.

%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="/A089462/b089462.txt">Table of n, a(n) for n = 0..385</a>

%F a(n) = Sum_{k=0..n} 2*(n-k+2)^(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+2)^(n-m-j+1)*(m+j)!/(-2)^j)/m!.

%F a(n) ~ 2 * exp(5/2) * n^(n-1). - _Vaclav Kotesovec_, Oct 11 2020

%t Table[Sum[Sum[Binomial[m, j]*Binomial[n, n - m - j + 1]*(n + 2)^(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+2)^(n-m-j+1)*(m+j)!/(-2)^j)/m!))

%Y Cf. A001858, A089460.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 05 2003

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Last modified June 14 01:29 EDT 2021. Contains 345016 sequences. (Running on oeis4.)