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Main diagonal of array A089944, in which the n-th row is the n-th binomial transform of the natural numbers.
4

%I #24 Sep 08 2022 08:45:12

%S 1,3,15,112,1125,14256,218491,3932160,81310473,1900000000,49516901511,

%T 1424099377152,44804009850925,1530735634132992,56439656982421875,

%U 2233785415175766016,94459960699823921169,4250383588380798812160,202774313738037680879743

%N Main diagonal of array A089944, in which the n-th row is the n-th binomial transform of the natural numbers.

%C a(n) is the number of labeled trees on n+1 nodes with a designated node or edge. - _Geoffrey Critzer_, Mar 25 2017

%H Alois P. Heinz, <a href="/A089945/b089945.txt">Table of n, a(n) for n = 0..386</a>

%F a(n) = (2*n+1)*(n+1)^(n-1).

%F E.g.f.: (-LambertW(-x)/x)*(1-LambertW(-x))/(1+LambertW(-x)).

%t nn = 30; T[z_] = -LambertW[-z]; Drop[Range[0, nn]! CoefficientList[Series[T[z] + T[z]^2/2, {z, 0, nn}], z], 1] (* _Geoffrey Critzer_, Mar 25 2017 *)

%t Table[(2 n + 1) (n + 1)^(n - 1), {n, 0, 18}] (* _Michael De Vlieger_, Mar 25 2017 *)

%o (PARI) a(n)=if(n<0,0,(2*n+1)*(n+1)^(n-1))

%o (Magma) [(2*n+1)*(n+1)^(n-1): n in [0..50]]; // _G. C. Greubel_, Nov 16 2017

%Y Cf. A089944, A089946, A245348.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 23 2003