A360193
a(n) = Sum_{k=0..n} (k-1)^(k-1) * binomial(n,k).
Original entry on oeis.org
-1, 0, 2, 9, 52, 445, 5166, 75019, 1300776, 26167257, 598577770, 15337224991, 435020120316, 13529095809541, 457727913937854, 16736043791509995, 657590281425958096, 27631245762003186865, 1236355641557737359570, 58689534518861119967287
Offset: 0
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a(n) = sum(k=0, n, (k-1)^(k-1)*binomial(n, k));
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-exp(x+lambertw(-x))))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(x*exp(x)/lambertw(-x)))
A105819
Triangle of the numbers of different forests of m rooted trees of smallest order 2, i.e., without isolated vertices, on N labeled nodes.
Original entry on oeis.org
0, 2, 0, 9, 0, 0, 64, 12, 0, 0, 625, 180, 0, 0, 0, 7776, 2730, 120, 0, 0, 0, 117649, 46410, 3780, 0, 0, 0, 0, 2097152, 893816, 99120, 1680, 0, 0, 0, 0, 43046721, 19389384, 2600640, 90720, 0, 0, 0, 0, 0, 1000000000, 469532790, 71734320, 3654000, 30240, 0
Offset: 1
a(8) = 12 because 4 vertices can be partitioned in two trees only in one way: both trees receiving 2 vertices. Two trees on 2 vertices can be labeled in binomial(4,2) ways and to each one of the 2*binomial(4,2) = 12 possibilities there are more 2 possible trees of order 2 in a forest. But since we have 2 trees of the same order, i.e., 2, we must divide 2*binomial(4,2)*2 by 2!.
Triangle T(n,k) begins:
: 0;
: 2, 0;
: 9, 0, 0;
: 64, 12, 0, 0;
: 625, 180, 0, 0, 0;
: 7776, 2730, 120, 0, 0, 0;
: 117649, 46410, 3780, 0, 0, 0, 0;
: 2097152, 893816, 99120, 1680, 0, 0, 0, 0;
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# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n=0,0,(n+1)^n), 9); # Peter Luschny, Jan 27 2016
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(
binomial(n-1, j-1)*j^(j-1)*x*b(n-j), j=2..n)))
end:
T:= (n, k)-> coeff(b(n), x, k):
seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Aug 13 2017
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BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
B = BellMatrix[Function[n, If[n == 0, 0, (n+1)^n]], rows];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)
A219034
Triangular array read by rows: T(n,k) is the number of forests of rooted trees on n labeled nodes with exactly k isolated nodes; n>=0, 0<=k<=n.
Original entry on oeis.org
1, 0, 1, 2, 0, 1, 9, 6, 0, 1, 76, 36, 12, 0, 1, 805, 380, 90, 20, 0, 1, 10626, 4830, 1140, 180, 30, 0, 1, 167839, 74382, 16905, 2660, 315, 42, 0, 1, 3091768, 1342712, 297528, 45080, 5320, 504, 56, 0, 1, 65127465, 27825912, 6042204, 892584, 101430, 9576, 756, 72, 0, 1
Offset: 0
Triangle T(n,k) begins:
1;
0, 1;
2, 0, 1;
9, 6, 0, 1;
76, 36, 12, 0, 1;
805, 380, 90, 20, 0, 1;
10626, 4830, 1140, 180, 30, 0, 1;
167839, 74382, 16905, 2660, 315, 42, 0, 1;
3091768, 1342712, 297528, 45080, 5320, 504, 56, 0, 1;
...
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b:= proc(n) option remember; expand(`if`(n=0, 1, add(i^(i-1)
*b(n-i)*binomial(n-1, i-1)*`if`(i=1, x, 1), i=1..n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..10); # Alois P. Heinz, Dec 31 2021
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nn=8; t=Sum[n^(n-1)x^n/n!, {n,1,nn}]; Range[0,nn]! CoefficientList[ Series[Exp[t-x+y x], {x,0,nn}], {x,y}] //Grid
A361916
a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * (k+1)^(k-1) / (k! * (n-2*k)!).
Original entry on oeis.org
1, 1, -1, -5, 25, 161, -1409, -12221, 158705, 1733185, -30136769, -397326709, 8696945929, 134416055905, -3555479651905, -63044502191789, 1957884163020001, 39178556553643649, -1398250387206450305, -31169265056007817445, 1257498026543130033401
Offset: 0
A361917
a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^k * (k+1)^(k-1) / (k! * (n-3*k)!).
Original entry on oeis.org
1, 1, 1, -5, -23, -59, 961, 7351, 29905, -877463, -9450719, -52724429, 2282907001, 31742360365, 225092745697, -12992587010129, -221436656404319, -1905297800257199, 137972958868569025, 2784953660339878507, 28177036295775415561, -2459373614334806266859
Offset: 0
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