A372315
Expansion of e.g.f. exp( x - LambertW(-2*x)/2 ).
Original entry on oeis.org
1, 2, 8, 68, 960, 18832, 471136, 14324480, 512733696, 21119803136, 984029612544, 51169331031040, 2937675286583296, 184560174104465408, 12594824112085327872, 927757127285523243008, 73369903633161123397632, 6200198958236463387836416
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-2*x)/2)))
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a(n) = sum(k=0, n, (2*k+1)^(k-1)*binomial(n, k));
A372316
Expansion of e.g.f. exp( x - LambertW(-3*x)/3 ).
Original entry on oeis.org
1, 2, 10, 125, 2644, 77597, 2904382, 132169403, 7083715240, 437031850841, 30506442905194, 2377038378159359, 204521399708464252, 19259006462435865413, 1970114326513629358654, 217556451608123850352523, 25794252755430105917806288, 3268152272130255473300883377
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-3*x)/3)))
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a(n) = sum(k=0, n, (3*k+1)^(k-1)*binomial(n, k));
A372320
Expansion of e.g.f. -exp( x + LambertW(-2*x)/2 ).
Original entry on oeis.org
-1, 0, 4, 36, 464, 8560, 206112, 6104896, 214376192, 8701657344, 400748710400, 20642974511104, 1175888936749056, 73389707156586496, 4980134850525986816, 365062349226075463680, 28747688571714736160768, 2420266280392895064506368
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(-exp(x+lambertw(-2*x)/2)))
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a(n) = sum(k=0, n, (2*k-1)^(k-1)*binomial(n, k));
A372321
Expansion of e.g.f. -exp( x + LambertW(-3*x)/3 ).
Original entry on oeis.org
-1, 0, 6, 81, 1620, 45765, 1671678, 74794671, 3958829640, 241898775273, 16756621904970, 1297547591499819, 111065107263415308, 10412999996499836541, 1061234184094567585326, 116812280111404106348415, 13810631408232372091755792, 1745470697932523785587735249
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(-exp(x+lambertw(-3*x)/3)))
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a(n) = sum(k=0, n, (3*k-1)^(k-1)*binomial(n, k));
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 *)
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