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A213262 Triangle read by rows: R*(n,k) (n>=2, k from 2 to n-1 (to 2 if n = 2), where R*(n,k) = number of trees with n nodes and k unlabeled end-nodes. 2
1, 1, 1, 1, 3, 2, 1, 12, 9, 3, 1, 60, 52, 18, 4, 1, 360, 360, 136, 30, 5, 1, 2520, 2880, 1205, 280, 45, 6, 1, 20160, 26040, 12090, 3025, 500, 63, 7, 1, 181440, 262080, 134610, 36546, 6375, 812, 84, 8, 1, 1814400, 2903040, 1641360, 484260, 90126, 11935, 1232, 108, 9, 1, 19958400, 35078400, 21712320, 6951840, 1386217, 193326, 20510, 1776, 135, 10, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,5

COMMENTS

All nodes are labeled except for the end-nodes.

LINKS

Table of n, a(n) for n=2..68.

F. Harary, A. Mowshowitz and J. Riordan, Labeled trees with unlabeled end-points, J. Combin. Theory, 6 (1969), 60-64.

EXAMPLE

Triangle begins:

[1],

[1],

[1, 1],

[3, 2, 1],

[12, 9, 3, 1],

[60, 52, 18, 4, 1],

[360, 360, 136, 30, 5, 1],

[2520, 2880, 1205, 280, 45, 6, 1],

[20160, 26040, 12090, 3025, 500, 63, 7, 1],

[181440, 262080, 134610, 36546, 6375, 812, 84, 8, 1],

[1814400, 2903040, 1641360, 484260, 90126, 11935, 1232, 108, 9, 1],

...

MAPLE

# This is for n >= 3:

with(combinat);

R:=proc(n, k) # This is for A151880

if n=1 then if k=1 then RETURN(1) else RETURN(0); fi

    elif (n=2 and k=2) then RETURN(1)

    elif (n=2 and k>2) then RETURN(0)

    else stirling2(n-2, n-k)*n!/k!;

    fi;

end;

Rstar:=proc(n, k)

if k=2 then

     if n <=4 then RETURN(1); else RETURN((n-2)!/2); fi;

else

   if k <= n-2 then add(binomial(n-i-1, k-i)*R(n-k, i), i=2..n-1);

   elif k=n-1 then 1;

   else 0;

   fi;

fi;

end;

g:=n->[seq(Rstar(n, k), k=2..n-1)];

[seq(g(n), n=3..16)];

MATHEMATICA

r[n_, k_] := Which[ n == 1, If[k == 1, Return[1], Return[0]], n == 2 && k == 2, Return[1], n == 2 && k > 2, Return[0], n > k > 0, StirlingS2[n-2, n-k]*n!/k!, True, 0]; rstar[n_, k_] := Which[ k == 2, If[ n <= 4 , Return[1], Return[(n-2)!/2]], k <= n-2, Sum[ Binomial[n-i-1, k-i]*r[n-k, i], {i, 2, n-1}] , k == n-1 , 1, True, 0]; g[n_] := Table[rstar[n, k], {k, 2, n-1}]; Join[{1}, Table[g[n], {n, 3, 13}] // Flatten] (* Jean-Fran├žois Alcover, Oct 05 2012, translated from Maple *)

CROSSREFS

Row sums give A001258. This is an improved version of A151880.

Sequence in context: A110616 A059418 A092582 * A280512 A068440 A246381

Adjacent sequences:  A213259 A213260 A213261 * A213263 A213264 A213265

KEYWORD

nonn,tabf

AUTHOR

N. J. A. Sloane, Jun 07 2012

STATUS

approved

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Last modified March 21 04:59 EDT 2019. Contains 321364 sequences. (Running on oeis4.)