A089945 Main diagonal of array A089944, in which the n-th row is the n-th binomial transform of the natural numbers.

1, 3, 15, 112, 1125, 14256, 218491, 3932160, 81310473, 1900000000, 49516901511, 1424099377152, 44804009850925, 1530735634132992, 56439656982421875, 2233785415175766016, 94459960699823921169, 4250383588380798812160, 202774313738037680879743

Offset: 0

Author

Paul D. Hanna, Nov 23 2003

Keywords

• nonn

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..386

Crossrefs

Cf. A089944, A089946, A245348.

Programs

Magma

[(2*n+1)*(n+1)^(n-1): n in [0..50]]; // G. C. Greubel, Nov 16 2017

Mathematica

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 *)
Table[(2 n + 1) (n + 1)^(n - 1), {n, 0, 18}] (* Michael De Vlieger, Mar 25 2017 *)

PARI

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

Formula

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

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