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A291559
Total height of all (unlabeled) rooted identity trees with n vertices.
2
0, 0, 1, 2, 5, 10, 23, 52, 120, 275, 644, 1508, 3558, 8418, 20012, 47699, 114082, 273476, 657250, 1582817, 3819514, 9233059, 22356918, 54216429, 131663670, 320158789, 779461271, 1899830067, 4635492672, 11321595218, 27677333555, 67720658475, 165835173692
OFFSET
0,4
LINKS
FORMULA
a(n) = Sum_{h=0..n-2} Sum_{t=1..n-1-h} (h+1) * A291529(n-1,h,t).
EXAMPLE
: a(5) = 10 = 4 + 3 + 3 : a(4) = 5 = 3 + 2 :
: : :
: o o o : o o :
: | | / \ : | / \ :
: o o o o : o o o :
: | / \ | : | | :
: o o o o : o o :
: | | | : | :
: o o o : o :
: | : :
: o : :
: : :
MAPLE
b:= proc(n, i, t, h) option remember; expand(`if`(n=0 or h=0 or i=1,
`if`(n<2, x^(t*n), 0), b(n, i-1, t, h)+add(x^(t*j)*binomial(
b(i-1$2, 0, h-1), j)*b(n-i*j, i-1, t, h), j=1..n/i)))
end:
g:= (n, h)-> b(n$2, 1, h)-`if`(h=0, 0, b(n$2, 1, h-1)):
F:= (n, h, t)-> coeff(g(n, h), x, t):
a:= n-> add(add((h+1)*F(n-1, h, t), t=1..n-1-h), h=0..n-2):
seq(a(n), n=0..37);
MATHEMATICA
b[n_, i_, t_, h_] := b[n, i, t, h] = Expand[If[n == 0 || h == 0 || i == 1,
If[n < 2, x^(t*n), 0], b[n, i-1, t, h] + Sum[x^(t*j)*Binomial[
b[i-1, i-1, 0, h-1], j]*b[n-i*j, i-1, t, h], {j, 1, n/i}]]];
g[n_, h_] := b[n, n, 1, h] - If[h == 0, 0, b[n, n, 1, h-1]];
F[n_, h_, t_] := Coefficient[g[n, h], x, t];
a[n_] := Sum[Sum[(h+1)*F[n-1, h, t], {t, 1, n-1-h}], {h, 0, n-2}];
Table[a[n], {n, 0, 37}] (* Jean-François Alcover, Dec 08 2023, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 26 2017
STATUS
approved