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%I #21 Sep 08 2022 08:44:35
%S 1,0,1,1,16,61,806,6329,89272,1082281,17596162,284074165,5407229972,
%T 107539072733,2380274168806,55833426732529,1418006883852784,
%U 38195636967960913,1097755724834189834,33345176998235584301,1071124330593423824908,36203857373308709200645
%N Number of edge-labeled series-reduced trees with n nodes.
%H G. C. Greubel, <a href="/A007831/b007831.txt">Table of n, a(n) for n = 1..400</a>
%H P. J. Cameron, <a href="https://doi.org/10.37236/1198">Counting two-graphs related to trees</a>, Elec. J. Combin., Vol. 2, #R4.
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F a(n) = A005512(n+1) / (n+1) for n >= 2. - _Sean A. Irvine_, Feb 03 2018
%F E.g.f.: 1/(2*x) + (x-1)/2 - ((1+x)/(2*x))*(1 + LambertW(-x/(1+x)))^2. - _G. C. Greubel_, Mar 08 2020
%p seq( `if`(n=1, 1, (n-1)!*add((-1)^k*binomial(n+1, k)*(n-k+1)^(n-k-1)/( (n+1)*(n-k-1)!), k = 0..n-1)), n=1..20); # _G. C. Greubel_, Mar 08 2020
%t Table[If[n==1, 1, (n-1)!*Sum[(-1)^k*Binomial[n+1,k]*(n-k+1)^(n-k-1)/((n+1)*(n - k-1)!), {k,0,n-1}]], {n, 20}] (* _G. C. Greubel_, Mar 08 2020 *)
%o (PARI) a(n) = if(n==1, 1, (n-1)!*sum(k=0, n-1, (-1)^k*binomial(n+1,k)*(n-k+1 )^(n-k-1)/( (n+1)*(n-k-1)!))); \\ _G. C. Greubel_, Mar 08 2020
%o (Magma) [1] cat [Factorial(n-1)*(&+[(-1)^k*Binomial(n+1,k)*(n-k+1)^(n-k-1)/((n+1)*Factorial(n-k-1)): k in [0..n-1]]): n in [2..20]] // _G. C. Greubel_, Mar 08 2020
%o (Sage) [1]+[factorial(n-1)*sum((-1)^k*binomial(n+1,k)*(n-k+1)^(n-k-1)/( (n+1)*factorial(n-k-1)) for k in (0..n-1)) for n in (2..20)] # _G. C. Greubel_, Mar 08 2020
%Y Cf. A005512.
%K nonn
%O 1,5
%A _Peter J. Cameron_
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