A245496 -id:A245496 - cp's OEIS frontend

A231797 Number of endofunctions on [n] such that no element has a preimage of cardinality one.

1, 0, 2, 3, 40, 205, 2556, 24409, 347712, 4794633, 81163900, 1434596581, 28725779952, 612610306477, 14280306222924, 354958921699425, 9471543095892736, 268347925179992593, 8075532017006497404, 256672899448317943453, 8603440900030816095600

Offset: 0

Alois P. Heinz, Nov 13 2013

nonn

			a(2) = 2: (1,1), (2,2).
a(3) = 3: (1,1,1), (2,2,2), (3,3,3).

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..410 (first 181 terms from Alois P. Heinz)
R. K. Guy, Letter to N. J. A. Sloane, Jun 21, 1975

Column k=0 of A206823.
A diagonal of A241580. Cf. also A241581.
Column k=1 of A245405.
Cf. A245496.

with(combinat):
b:= proc(t, i, u) option remember; `if`(t=0, 1,
      `if`(i<2, 0, b(t, i-1, u) +add(multinomial(t, t-i*j, i$j)
      *b(t-i*j, i-1, u-j)*u!/(u-j)!/j!, j=1..t/i)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..25);

multinomial[n_, k_List] := n!/Times @@ (k!); b[t_, i_, u_] := b[t, i, u] = If[t == 0, 1, If[i < 2, 0, b[t, i - 1, u] + Sum[multinomial[t, Join[{ t - i*j}, Array[i &, j]]] * b[t - i*j, i - 1, u - j]*u!/(u - j)!/j!, {j, 1, t/i}]]]; a[n_] := b[n, n, n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 27 2013, translated from Maple *)
Table[n!*SeriesCoefficient[(E^x-x)^n,{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 23 2014 *)
Flatten[{1,Table[(-1)^n*n! + Sum[Binomial[n,j]^2*(-1)^j*(n-j)^(n-j)*j!,{j,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, Jul 25 2014 *)

{a(n) = n! * polcoeff((exp(x + x*O(x^n)) - x)^n, n)}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Jan 30 2015

a(n) = n! * [x^n] (exp(x)-x)^n.
a(n) ~ (1-exp(-1))^(n+1/2) * n^n. - Vaclav Kotesovec, Jul 23 2014
E.g.f.: 1 / ((1 + x) * (1 + LambertW(-x/(1 + x)))). - Ilya Gutkovskiy, Jan 25 2020
a(n) = Sum_{k=0..n} (-1)^(n-k)*(n-k)!*k^k*binomial(n,k)^2. - Ridouane Oudra, Jul 14 2025

A245493 a(n) = n! * [x^n] (exp(x)+x^2/2!)^n.

Original entry on oeis.org

1, 1, 6, 45, 508, 7225, 126306, 2606065, 62075952, 1675774089, 50565938050, 1686510607111, 61609858744248, 2446470026497705, 104922088624078194, 4833250468667819325, 238004208840601580416, 12476420334546637657489, 693675026024580055139778

Offset: 0

Vaclav Kotesovec, Jul 24 2014

In general, if a(n) = n! * [x^n] (exp(x) + x^k/k!)^n, k>=1, then limit n-> infinity (a(n)/n!)^(1/n) = ((1-k*r)/(1-r))^(k-1) / (r*k!), where r is the root of the equation exp((k*r-1)/(1-r)) = r*k! * (1-r)^(k-1) / (1-k*r)^k.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A245406, A245405, A245496.

Table[n!*SeriesCoefficient[(E^x + x^2/2)^n, {x, 0, n}], {n, 0, 20}]
With[{k=2}, Flatten[{1, Table[Sum[Binomial[n, j]*Binomial[n, k*j]*(n-j)^(n-k*j)*(k*j)!/(k!)^j, {j, 0, n/k}], {n, 1, 20}]}]]

a(n) ~ c * d^n * n^n / exp(n), where d = (1-2*r)/(2*r*(1-r)) = 3.177499696443893762475339445134038..., where r = 0.13317988718414524112... is the root of the equation exp((2*r-1)/(1-r)) = 2*r*(1-r)/(1-2*r)^2, and c = 1.061620103934913384222610538939... .

A331726 E.g.f.: -LambertW(-x/(1 - x)) / (1 - x).

Original entry on oeis.org

0, 1, 6, 45, 448, 5825, 95796, 1926043, 45944256, 1269187137, 39840825700, 1400286658331, 54462564354672, 2321934762267601, 107664031299459012, 5393893268767761675, 290341440380472614656, 16710435419661861992705, 1024009456958258244673860

Offset: 0

Ilya Gutkovskiy, Jan 25 2020

nonn

Cf. A000169, A052871, A245496, A277505, A331727.

nmax = 18; CoefficientList[Series[-LambertW[-x/(1 - x)]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k]^2 k! (n - k)^(n - k - 1), {k, 0, n - 1}], {n, 0, 18}]

seq(n)={Vec(serlaplace(-lambertw(-x/(1 - x) + O(x*x^n)) / (1 - x)), -(n+1))} \\ Andrew Howroyd, Jan 25 2020

a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * k! * (n - k)^(n - k - 1).

a(n) ~ (1 + exp(-1))^(n + 3/2) * n^(n-1). - Vaclav Kotesovec, Jan 26 2020

cp's OEIS Frontend

A231797 Number of endofunctions on [n] such that no element has a preimage of cardinality one.

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A245493 a(n) = n! * [x^n] (exp(x)+x^2/2!)^n.

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A331726 E.g.f.: -LambertW(-x/(1 - x)) / (1 - x).

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