

A089945


Main diagonal of array A089944, in which the nth row is the nth binomial transform of the natural numbers.


4



1, 3, 15, 112, 1125, 14256, 218491, 3932160, 81310473, 1900000000, 49516901511, 1424099377152, 44804009850925, 1530735634132992, 56439656982421875, 2233785415175766016, 94459960699823921169, 4250383588380798812160, 202774313738037680879743
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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)^(n1).
E.g.f.: (LambertW(x)/x)*(1LambertW(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)^(n1))
(MAGMA) [(2*n+1)*(n+1)^(n1): 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



