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A007831
Number of edge-labeled series-reduced trees with n nodes.
3
1, 0, 1, 1, 16, 61, 806, 6329, 89272, 1082281, 17596162, 284074165, 5407229972, 107539072733, 2380274168806, 55833426732529, 1418006883852784, 38195636967960913, 1097755724834189834, 33345176998235584301, 1071124330593423824908, 36203857373308709200645
OFFSET
1,5
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
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 *)
PROG
(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
(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
(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
CROSSREFS
Cf. A005512.
Sequence in context: A241523 A264632 A373667 * A214524 A118254 A356249
KEYWORD
nonn
STATUS
approved