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

0,2

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

Formula

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

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

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 *)

Prog

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

Crossrefs

Cf. A089944, A089946, A245348.

Sequence in context: A001063 A130168 A267083 * A135083 A323772 A058104

Adjacent sequences: A089942 A089943 A089944 * A089946 A089947 A089948

Keyword

nonn

Author

Paul D. Hanna, Nov 23 2003

Status

approved