A007831 Number of edge-labeled series-reduced trees with n nodes.

1, 0, 1, 1, 16, 61, 806, 6329, 89272, 1082281, 17596162, 284074165, 5407229972, 107539072733, 2380274168806, 55833426732529, 1418006883852784, 38195636967960913, 1097755724834189834, 33345176998235584301, 1071124330593423824908, 36203857373308709200645

Offset: 1

Author

Peter J. Cameron

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..400
• P. J. Cameron, Counting two-graphs related to trees, Elec. J. Combin., Vol. 2, #R4.
• Index entries for sequences related to trees

Crossrefs

Cf. A005512.

Programs

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

Maple

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

Mathematica

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 *)

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

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

Formula

a(n) = A005512(n+1) / (n+1) for n >= 2. - Sean A. Irvine, Feb 03 2018

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