A069856
E.g.f.: exp(x)/(1+LambertW(x)).
Original entry on oeis.org
1, 0, 3, -17, 169, -2079, 31261, -554483, 11336753, -262517615, 6791005621, -194103134499, 6074821125385, -206616861429575, 7588549099814957, -299320105069298459, 12619329503201165281, -566312032570838608863, 26952678355224681891685
Offset: 0
Joe Keane (jgk(AT)jgk.org), May 03 2002
- sci.math article 3CBC2B66.224E(AT)olympus.mons
-
t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Range[0, 20]! CoefficientList[Series[Exp[-x]/(1 - t), {x, 0, 20}], x] (* Geoffrey Critzer, Nov 13 2011 *)
Range[0, 18]! CoefficientList[ Series[ Exp[x]/(1 + LambertW[x]), {x, 0, 18}], x] (* Robert G. Wilson v, Nov 28 2012 *)
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my(x='x+O('x^20)); Vec(serlaplace(exp(x)/(1+lambertw(x)))) \\ G. C. Greubel, Jun 11 2017
A349454
Number T(n,k) of endofunctions on [n] with exactly k fixed points, all of which are isolated; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 1, 0, 1, 8, 3, 0, 1, 81, 32, 6, 0, 1, 1024, 405, 80, 10, 0, 1, 15625, 6144, 1215, 160, 15, 0, 1, 279936, 109375, 21504, 2835, 280, 21, 0, 1, 5764801, 2239488, 437500, 57344, 5670, 448, 28, 0, 1, 134217728, 51883209, 10077696, 1312500, 129024, 10206, 672, 36, 0, 1
Offset: 0
Triangle T(n,k) begins:
1;
0, 1;
1, 0, 1;
8, 3, 0, 1;
81, 32, 6, 0, 1;
1024, 405, 80, 10, 0, 1;
15625, 6144, 1215, 160, 15, 0, 1;
279936, 109375, 21504, 2835, 280, 21, 0, 1;
5764801, 2239488, 437500, 57344, 5670, 448, 28, 0, 1;
...
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T:= (n, k)-> binomial(n, k)*(n-k-1)^(n-k):
seq(seq(T(n, k), k=0..n), n=0..10);
A350134
Number of endofunctions on [n] with at least one isolated fixed point.
Original entry on oeis.org
0, 1, 1, 10, 87, 1046, 15395, 269060, 5440463, 124902874, 3208994379, 91208536112, 2841279322871, 96258245162678, 3523457725743059, 138573785311560916, 5827414570508386335, 260928229315498155314, 12393729720071855683739, 622422708333615857463608
Offset: 0
a(3) = 10: 123, 122, 133, 132, 121, 323, 321, 113, 223, 213.
-
g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, t) option remember; `if`(n=0, t, add(g(i)*
b(n-i, `if`(i=1, 1, t))*binomial(n-1, i-1), i=1..n))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);
-
g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, t_] := b[n, t] = If[n == 0, t, Sum[g[i]*
b[n - i, If[i == 1, 1, t]]*Binomial[n - 1, i - 1], {i, 1, n}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 27 2022, after Alois P. Heinz *)
A348590
Number of endofunctions on [n] with exactly one isolated fixed point.
Original entry on oeis.org
0, 1, 0, 9, 68, 845, 12474, 218827, 4435864, 102030777, 2625176150, 74701061831, 2329237613988, 78972674630005, 2892636060014050, 113828236497224355, 4789121681108775344, 214528601554419809777, 10193616586275094959534, 512100888749268955942015
Offset: 0
a(3) = 9: 122, 133, 132, 121, 323, 321, 113, 223, 213.
-
g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, t) option remember; `if`(n=0, t, add(g(i)*
b(n-i, `if`(i=1, 1, t))*binomial(n-1, i-1), i=1+t..n))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);
-
g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}] ;
b[n_, t_] := b[n, t] = If[n == 0, t, Sum[g[i]*
b[n - i, If[i == 1, 1, t]]*Binomial[n - 1, i - 1], {i, 1 + t, n}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, May 16 2022, after Alois P. Heinz *)
Showing 1-4 of 4 results.
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