A151880
Triangle: R*(n,k) (n>=2, k from 2 to n-1 or to 2 if n = 2), where R*(n,k) = number of trees with n nodes and k unlabeled end-nodes.
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 3, 1, 3, 9, 12, 1, 4, 18, 52, 60, 1, 5, 30, 136, 360, 360
Offset: 0
Triangle (in fact the columns in the original have been reversed and the triangle transposed):
(n=2) 1
(n=3) 1
(n=4) 1 1
(n=5) 1 2 3
(n=6) 1 3 9 12
(n=7) 1 4 18 52 60
(n=8) 1 5 30 136 360 360
See
A213262 for a better version with more terms and a program.
A213262
Triangle read by rows: R*(n,k) (n>=2, k from 2 to n-1 or to 2 if n = 2), where R*(n,k) = number of trees with n nodes and k unlabeled end-nodes.
Original entry on oeis.org
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
Offset: 2
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],
...
-
# 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)];
-
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-François Alcover, Oct 05 2012, translated from Maple *)
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