A069856 -id:A069856 - cp's OEIS frontend

A086331 Expansion of e.g.f. exp(x)/(1 + LambertW(-x)).

1, 2, 7, 43, 393, 4721, 69853, 1225757, 24866481, 572410513, 14738647221, 419682895325, 13094075689225, 444198818128313, 16278315877572141, 640854237634448101, 26973655480577228769, 1208724395795734172705, 57453178877303382607717, 2887169565412587866031533

Offset: 0

Vladeta Jovovic, Sep 01 2003

nonn

Binomial transform of A000312. - Tilman Neumann, Dec 13 2008

a(n) is the number of partial functions on {1,2,...,n} that are endofunctions. See comments in A000169 and A126285 by Franklin T. Adams-Watters. - Geoffrey Critzer, Dec 19 2011

			a(2) = 7 because {}->{}, 1->1, 2->2, and the four functions from {1,2} into {1,2}. Note A000169(2) = 9 because it counts these 7 and 1->2, 2->1.

Winston de Greef, Table of n, a(n) for n = 0..385 (first 201 terms from Vincenzo Librandi)
V. Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013

Cf. A069856, A204042, A277454, A277456, A323280.

a:= n-> add(binomial(n,k)*k^k, k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 30 2021

nn=10;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[Series[Exp[x]/(1-t),{x,0,nn}],x]  (* Geoffrey Critzer, Dec 19 2011 *)

a(n) = sum(k=0,n, binomial(n, k)*k^k ); \\ Joerg Arndt, May 10 2013

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-x)^(k+1))) \\ Seiichi Manyama, Jul 04 2022

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=0, N, (k*x)^k/k!))) \\ Seiichi Manyama, Jul 04 2022

a(n) = Sum_{k=0..n} binomial(n,k)*k^k.
a(n) ~ e^(1/e)*n^n * (1 + 1/(2*e*n)) ~ 1.444667861... * n^n. - Vaclav Kotesovec, Nov 27 2012
G.f.: Sum_{k>=0} (k * x)^k/(1 - x)^(k+1). - Seiichi Manyama, Jul 04 2022

A003725 E.g.f.: exp( x * exp(-x) ).

Original entry on oeis.org

1, 1, -1, -2, 9, -4, -95, 414, 49, -10088, 55521, -13870, -2024759, 15787188, -28612415, -616876274, 7476967905, -32522642896, -209513308607, 4924388011050, -38993940088199, 11731860520780, 3807154270837281

Offset: 0

R. H. Hardin

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..557
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3.

Cf. A000248, A069856, A072034. A215652.

Cf. this sequence (k=1), A292909 (k=2), A292910 (k=3), A292912 (k=4).

With[{nn=30},CoefficientList[Series[Exp[x Exp[-x]],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Oct 20 2011 *)

Vec(serlaplace(exp(exp(-x) * x))) \\ Charles R Greathouse IV, Sep 26 2017

a(n) = Sum_{k=0..n} (-k)^(n-k)*binomial(n, k). - Vladeta Jovovic, Mar 15 2003
First column of A215652. - Peter Bala, Sep 14 2012
G.f.: Sum_{k>=0} x^k/(1 + k*x)^(k+1). - Ilya Gutkovskiy, Jun 25 2018

A277473 E.g.f.: -exp(x)*LambertW(-x).

Original entry on oeis.org

0, 1, 4, 18, 116, 1060, 12702, 187810, 3296120, 66897288, 1540762010, 39693752494, 1130866726596, 35300006582620, 1198036854980630, 43921652697277170, 1729775120233353968, 72831210167041246480, 3264674481128340280242, 155220967397580333229270

Offset: 0

Vaclav Kotesovec, Oct 17 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000169, A069856, A086331, A277474.

Partial sums of A038051.

CoefficientList[Series[-Exp[x]*LambertW[-x], {x, 0, 20}], x] * Range[0, 20]!
Table[Sum[Binomial[n, k]*k^(k-1), {k, 1, n}], {n, 0, 20}]

x='x+O('x^50); concat([0], Vec(serlaplace(-exp(x)*lambertw(-x)))) \\ G. C. Greubel, Jun 11 2017

a(n) = Sum_{k=1..n} binomial(n,k) * k^(k-1).

a(n) ~ exp(exp(-1)) * n^(n-1).

A277464 Expansion of e.g.f. cosh(x)/(1 + LambertW(-x)).

Original entry on oeis.org

1, 1, 5, 30, 281, 3400, 50557, 890120, 18101617, 417464064, 10764826421, 306893014912, 9584448407305, 325407839778944, 11933432488693549, 470087171351873280, 19796492491889197025, 887518214183286390784, 42202928616264032249701, 2121583256369642798845952

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000312, A069856, A086331, A277462, A277463.

CoefficientList[Series[Cosh[x]/(1+LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]!
Table[(1+(-1)^n + Sum[(1+(-1)^(n-k)) * Binomial[n,k] * k^k, {k, 1, n}])/2, {n, 0, 25}]

x='x+O('x^50); Vec(serlaplace(cosh(x)/(1 + lambertw(-x)))) \\ G. C. Greubel, Nov 07 2017

a(n) = sum(k=0, n\2, (n-2*k)^(n-2*k)*binomial(n, 2*k)); \\ Seiichi Manyama, Feb 15 2023

a(n) ~ cosh(exp(-1)) * n^n.

a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^(n-2*k) * binomial(n,2*k). - Seiichi Manyama, Feb 15 2023

A277463 E.g.f.: sinh(x)/(1+LambertW(-x)).

Original entry on oeis.org

0, 1, 2, 13, 112, 1321, 19296, 335637, 6764864, 154946449, 3973820800, 112789880413, 3509627281920, 118790978349369, 4344883388878592, 170767066282574821, 7177162988688031744, 321206181612447781921, 15250250261039350358016, 765586309042945067185581

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000312, A069856, A086331, A277461, A277464.

CoefficientList[Series[Sinh[x]/(1+LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]!
Table[(1-(-1)^n + Sum[(1-(-1)^(n-k)) * Binomial[n,k] * k^k, {k, 1, n}])/2, {n, 0, 25}]

x='x+O('x^50); concat([0], Vec(serlaplace(sinh(x)/(1 + lambertw(-x))))) \\ G. C. Greubel, Nov 05 2017

a(n) ~ sinh(exp(-1)) * n^n.

A290158 a(n) = n! * [x^n] exp(-n*x)/(1 + LambertW(-x)).

Original entry on oeis.org

1, 0, 4, -9, 208, -1525, 33516, -463099, 11293248, -231839577, 6517863100, -175791146311, 5723314711632, -189288946716181, 7083626583237036, -275649085963046475, 11724766124450058496, -522717581675749841713, 24981438186138642481404

Offset: 0

Ilya Gutkovskiy, Oct 06 2017

sign

The n-th term of the n-th inverse binomial transform of A000312.

G. C. Greubel, Table of n, a(n) for n = 0..385
N. J. A. Sloane, Transforms

Cf. A000312, A069856, A290840, A318615, A356806.

Table[n! SeriesCoefficient[Exp[-n x]/(1 + LambertW[-x]), {x, 0, n}], {n, 0, 18}]

a(n) = (-1)^n*n!*sum(k=0, n\2, n^k*stirling(n-k, k, 2)/(n-k)!); \\ Seiichi Manyama, May 05 2023

a(n) ~ (-1)^n * n^n / (1 + LambertW(1)). - Vaclav Kotesovec, Oct 06 2017
From Seiichi Manyama, May 05 2023: (Start)
a(n) = (-1)^n * n! * [x^n] exp(n * x * (exp(x) - 1)).
a(n) = (-1)^n * n! * Sum_{k=0..floor(n/2)} n^k * Stirling2(n-k,k)/(n-k)!.
a(n) = [x^n] Sum_{k>=0} (k*x)^k / (1 + n*x)^(k+1).
a(n) = Sum_{k=0..n} (-n)^(n-k) * k^k * binomial(n,k). (End)

A350212 Number T(n,k) of endofunctions on [n] with exactly k isolated fixed points; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 3, 0, 1, 17, 9, 0, 1, 169, 68, 18, 0, 1, 2079, 845, 170, 30, 0, 1, 31261, 12474, 2535, 340, 45, 0, 1, 554483, 218827, 43659, 5915, 595, 63, 0, 1, 11336753, 4435864, 875308, 116424, 11830, 952, 84, 0, 1, 262517615, 102030777, 19961388, 2625924, 261954, 21294, 1428, 108, 0, 1

Offset: 0

Alois P. Heinz, Dec 19 2021

			T(3,1) = 9: 122, 133, 132, 121, 323, 321, 113, 223, 213.
Triangle T(n,k) begins:
         1;
         0,       1;
         3,       0,      1;
        17,       9,      0,      1;
       169,      68,     18,      0,     1;
      2079,     845,    170,     30,     0,   1;
     31261,   12474,   2535,    340,    45,   0,  1;
    554483,  218827,  43659,   5915,   595,  63,  0, 1;
  11336753, 4435864, 875308, 116424, 11830, 952, 84, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-1 give: |A069856|, A348590.
Row sums give A000312.
T(n+1,n-1) gives A045943.
Cf. A001865, A008290, A008291, A055134, A055897, A060281, A086659, A124323, A350134, A349454.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, m) option remember; `if`(n=0, x^m, add(g(i)*
      b(n-i, m+`if`(i=1, 1, 0))*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..10);
# second Maple program:
A350212 := (n,k)-> add((-1)^(j-k)*binomial(j,k)*binomial(n,j)*(n-j)^(n-j), j=0..n):
seq(print(seq(A350212(n, k), k=0..n)), n=0..9); # Mélika Tebni, Nov 24 2022

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, m_] := b[n, m] = If[n == 0, x^m, Sum[g[i]*
     b[n - i, m + If[i == 1, 1, 0]]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 11 2022, after Alois P. Heinz *)

Sum_{k=0..n} k * T(n,k) = A055897(n).
Sum_{k=1..n} T(n,k) = A350134(n).
From Mélika Tebni, Nov 24 2022: (Start)
T(n,k) = binomial(n, k)*|A069856(n-k)|.
E.g.f. column k: exp(-x)*x^k / ((1 + LambertW(-x))*k!).
T(n,k) = Sum_{j=0..n} (-1)^(j-k)*binomial(j, k)*binomial(n, j)*(n-j)^(n-j). (End)

A362019 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = (-1)^n * Sum_{j=0..n} (-kj)^j binomial(n,j).

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, 3, -1, 1, 2, 13, 17, 1, 1, 3, 31, 173, 169, -1, 1, 4, 57, 629, 3321, 2079, 1, 1, 5, 91, 1547, 18025, 81529, 31261, -1, 1, 6, 133, 3089, 58993, 662639, 2443333, 554483, 1, 1, 7, 183, 5417, 147081, 2888979, 29752957, 86475493, 11336753, -1

Offset: 0

Seiichi Manyama, May 05 2023

			Square array begins:
   1,    1,     1,      1,       1,       1, ...
  -1,    0,     1,      2,       3,       4, ...
   1,    3,    13,     31,      57,      91, ...
  -1,   17,   173,    629,    1547,    3089, ...
   1,  169,  3321,  18025,   58993,  147081, ...
  -1, 2079, 81529, 662639, 2888979, 8998399, ...

Eric Weisstein's World of Mathematics, Lambert W-Function.

Columns k=0..3 give A033999, (-1)^n * A069856(n), A362859, A362860.
Main diagonal gives A362862.
Cf. A362856.

T(n, k) = (-1)^n*sum(j=0, n, (-k*j)^j*binomial(n, j));

E.g.f. of column k: exp(-x) / (1 + LambertW(-k*x)).

G.f. of column k: Sum_{j>=0} (k*j*x)^j / (1 + x)^(j+1).

A277474 E.g.f.: -exp(-x)*LambertW(-x).

Original entry on oeis.org

0, 1, 0, 6, 36, 380, 4830, 74382, 1342712, 27825912, 651274650, 16994464850, 489240628932, 15404364096420, 526634857318934, 19428038813967630, 769280055136105200, 32543192449030871792, 1464827827285673677746, 69903432558329996409642, 3525344776953738276010940

Offset: 0

Vaclav Kotesovec, Oct 17 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000169, A069856, A086331, A277473.

CoefficientList[Series[-Exp[-x]*LambertW[-x], {x, 0, 20}], x] * Range[0, 20]!
Table[Sum[(-1)^(n-k)*Binomial[n, k]*k^(k-1), {k, 1, n}], {n, 0, 20}]

x='x+O('x^50); concat([0], Vec(serlaplace(-exp(-x)*lambertw(-x)))) \\ G. C. Greubel, Nov 07 2017

a(n) = Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * k^(k-1).

a(n) ~ exp(-exp(-1)) * n^(n-1).

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

Alois P. Heinz, Dec 15 2021

nonn

			a(3) = 10: 123, 122, 133, 132, 121, 323, 321, 113, 223, 213.

Alois P. Heinz, Table of n, a(n) for n = 0..386

Column k=1 of A347999.

Cf. A000312, A001865, A045531, A069856, A204042, A350212.

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 *)

a(n) = A000312(n) - abs(A069856(n)).

a(n) = Sum_{k=1..n} A350212(n,k).

cp's OEIS Frontend

A086331 Expansion of e.g.f. exp(x)/(1 + LambertW(-x)).

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A003725 E.g.f.: exp( x * exp(-x) ).

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A277473 E.g.f.: -exp(x)*LambertW(-x).

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A277464 Expansion of e.g.f. cosh(x)/(1 + LambertW(-x)).

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A277463 E.g.f.: sinh(x)/(1+LambertW(-x)).

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A290158 a(n) = n! * [x^n] exp(-n*x)/(1 + LambertW(-x)).

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A350212 Number T(n,k) of endofunctions on [n] with exactly k isolated fixed points; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

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A362019 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = (-1)^n * Sum_{j=0..n} (-kj)^j binomial(n,j).