A277506 -id:A277506 - cp's OEIS frontend

A354436 a(n) = n! * Sum_{k=0..n} k^(n-k)/k!.

1, 1, 3, 13, 85, 801, 10231, 168253, 3437673, 85162465, 2511412651, 86805640461, 3469622549053, 158523442439233, 8198514736542495, 476003264246418301, 30804251925861439441, 2207978115389469465153, 174304316334466458575443

Offset: 0

Seiichi Manyama, May 28 2022

nonn

Cf. A006153, A026898, A010844, A277452, A277506, A354437.

Join[{1}, Table[n!*Sum[k^(n-k)/k!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 28 2022 *)

PARI
```
a(n) = n!*sum(k=0, n, k^(n-k)/k!);
```

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k*x)))))

from math import factorial
def A354436(n): return sum(factorial(n)*k**(n-k)//factorial(k) for k in range(n+1)) # Chai Wah Wu, May 28 2022

E.g.f.: Sum_{k>=0} x^k / (k! * (1 - k*x)).
a(n) ~ sqrt(Pi) * exp((2*n-1)/(2*LambertW(exp(1/2)*(2*n-1)/4)) - 2*n) * n^(2*n + 1/2) / (sqrt(1 + LambertW(exp(1/2)*(2*n-1)/4)) * 2^n * LambertW(exp(1/2)*(2*n-1)/4)^n). - Vaclav Kotesovec, May 28 2022
a(n) = Sum_{k=0..n} (n-k)^k*k!*binomial(n,k). - Ridouane Oudra, Jun 17 2025

A277509 Expansion of e.g.f. 1/((1+LambertW(-x))*(1+x)).

Original entry on oeis.org

1, 0, 4, 15, 196, 2145, 33786, 587041, 12080888, 278692497, 7213075030, 205967845281, 6444486304884, 219096784628761, 8044651840755362, 317224112769528945, 13371158269397088496, 599930571306586259745, 28547657791777984900014, 1436014157616531876023713

Offset: 0

Vaclav Kotesovec, Oct 18 2016

G. C. Greubel, Table of n, a(n) for n = 0..385
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A000312, A277506, A277508.

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

my(x='x+O('x^50)); Vec(serlaplace(1/((1 + lambertw(-x))*(1+x)))) \\ G. C. Greubel, Nov 12 2017

For n > 0, a(n) = (-1)^n*n!+Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * k^k * (n-k)!.
a(n) ~ n^n / (1+exp(-1)).
a(0) = 1; a(n) = -n*a(n-1) + n^n. - Seiichi Manyama, May 01 2023

A102743 Expansion of e.g.f. LambertW(-x)/(x*(x-1)).

Original entry on oeis.org

1, 2, 7, 37, 273, 2661, 32773, 491555, 8715409, 178438681, 4142334501, 107483043735, 3081956918857, 96759352320437, 3300826000845493, 121569984050610331, 4807542796319581089, 203167758634027130289

Offset: 0

Vladeta Jovovic, Feb 08 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A277506.

CoefficientList[Series[LambertW[-x]/(x*(x-1)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)

my(x='x+O('x^50)); Vec(serlaplace(lambertw(-x)/(x*(x-1)))) \\ G. C. Greubel, Nov 08 2017

a(n) = n!*Sum_{k=1..n+1} k^(k-1)/k!. - Vladeta Jovovic, Oct 17 2007
a(n) ~ exp(2)/(exp(1)-1) * n^(n-1). - Vaclav Kotesovec, Nov 27 2012
E.g.f.: W(0)/(2-2*x) , where W(k) = 1 + 1/( 1 - x*(k+2)^k/( x*(k+2)^k + (k+1)^k/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013
From Seiichi Manyama, May 01 2023: (Start)
E.g.f.: exp(-LambertW(-x))/(1-x).
a(0) = 1; a(n) = n*a(n-1) + (n+1)^(n-1). (End)

A277507 E.g.f.: -1/((1-LambertW(-x))*(1-x)).

Original entry on oeis.org

-1, 0, 0, 3, 28, 305, 3846, 57337, 998600, 20036529, 456403690, 11647754921, 329290975212, 10214585950153, 344897398385918, 12590837785019145, 494101941398352016, 20740772742716097377, 927276395603713539282, 43987299891665164562377, 2206610456287703987567540

Offset: 0

Vaclav Kotesovec, Oct 18 2016

sign

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

Cf. A277506, A277508.

CoefficientList[Series[-1/(1-LambertW[-x])/(1-x), {x, 0, 20}], x] * Range[0, 20]!

x='x+O('x^50); Vec(serlaplace(-1/((1 - lambertw(-x))*(1-x)))) \\ G. C. Greubel, Nov 08 2017

a(n) ~ n^(n-1) / (4*(1-exp(-1))).

A344229 a(n) = n*a(n-1) + n^signum(n mod 4), a(0) = 1.

Original entry on oeis.org

1, 2, 6, 21, 85, 430, 2586, 18109, 144873, 1303866, 13038670, 143425381, 1721104573, 22374359462, 313241032482, 4698615487245, 75177847795921, 1278023412530674, 23004421425552150, 437084007085490869, 8741680141709817381, 183575282975906165022

Offset: 0

Alois P. Heinz, May 12 2021

nonn

This sequence is one of many possible solutions to puzzle 16 on the Meerdaelquiz puzzle page, cf. the Delestinne link and A090805.

Alois P. Heinz, Table of n, a(n) for n = 0..450
Hugo Delestinne, Meerdaelquiz [broken link]
N. J. A. Sloane and Brady Haran, A Sequence with a Mistake, Numberphile video (2021)

Cf. A000142, A000522, A010873, A033540, A053817, A057427, A090805, A166486, A277506.

a:= proc(n) a(n):= n*a(n-1) + n^signum(n mod 4) end: a(0):= 1:
seq(a(n), n=0..23);

A300519 Convolution of n! and n^n.

Original entry on oeis.org

1, 2, 7, 39, 321, 3603, 51391, 884873, 17770445, 406673247, 10431884283, 296262164637, 9224841015745, 312441152401067, 11434829066996087, 449675059390576257, 18908960744072894325, 846638474386244188311, 40213487658138717885907, 2019543479160709325145893

Offset: 0

Vaclav Kotesovec, Mar 08 2018

nonn

Cf. A000142, A000312, A061711, A277506.

Cf. A107713, A108953, A110467.

Table[Sum[If[k == 0, 1, k^k] * (n-k)!, {k, 0, n}], {n, 0, 20}]

a(n) = Sum_{k=0..n} k^k * (n-k)!.

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

A356691 a(n) = n! * Sum_{k=0..n} k^(2*k)/k!.

Original entry on oeis.org

1, 2, 20, 789, 68692, 10109085, 2237436846, 693885130771, 287026057756824, 152677869816810537, 101526778698168105370, 82519543952519610272391, 80487081730821079456710228, 92779662255769290691336848973, 124775610962828705895908497741878

Offset: 0

Seiichi Manyama, Aug 23 2022

nonn

Cf. A062206, A277506, A350008, A356689.

PARI
```
a(n) = n!*sum(k=0, n, k^(2*k)/k!);
```

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+i^(2*i)); v;

a(0) = 1; a(n) = n*a(n-1) + n^(2*n).

A360596 Expansion of e.g.f. 1/( (1 - x) * (1 + LambertW(-2*x)) ).

Original entry on oeis.org

1, 3, 22, 282, 5224, 126120, 3742704, 131612432, 5347866752, 246490091136, 12704900911360, 724072211436288, 45209213973292032, 3068872654856532992, 225023336997933996032, 17724257054969009940480, 1492513932494133333753856, 133800772458366199028023296

Offset: 0

Seiichi Manyama, Feb 13 2023

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

Cf. A062971, A277506, A277509.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/((1-x)*(1+lambertw(-2*x)))))

PARI
```
a(n) = n!*sum(k=0, n, (2*k)^k/k!);
```

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+(2*i)^i); v;

a(n) = n! * Sum_{k=0..n} (2*k)^k / k!.
a(0)=1; a(n) = n*a(n-1) + (2*n)^n.
a(n) ~ 2^(n+1) * n^n / (2 - exp(-1)). - Vaclav Kotesovec, Feb 13 2023

A374844 a(n) = n! * Sum_{k=1..n} k^k / k!.

Original entry on oeis.org

0, 1, 6, 45, 436, 5305, 78486, 1372945, 27760776, 637267473, 16372674730, 465411092641, 14501033559948, 491388542871577, 17991446425760094, 707765586767260785, 29770993461985724176, 1333347150740094075169, 63346656788618230928466

Offset: 0

Seiichi Manyama, Jul 22 2024

nonn

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

Cf. A007526, A030297, A337001, A337002.

Cf. A000142, A277506.

a:= proc(n) a(n):= n*a(n-1) + n^n end: a(0):= 0:
seq(a(n), n=0..23);  # Alois P. Heinz, Jul 22 2024

PARI
```
a(n) = n!*sum(k=1, n, k^k/k!);
```

a(0) = 0; a(n) = n*a(n-1) + n^n.
a(n) = A277506(n) - n!.
E.g.f.: -1/( (1 + 1/LambertW(-x)) * (1 - x) ).
a(n) ~ n^n / (1 - exp(-1)). - Vaclav Kotesovec, Jul 22 2024

cp's OEIS Frontend

A354436 a(n) = n! * Sum_{k=0..n} k^(n-k)/k!.

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A277509 Expansion of e.g.f. 1/((1+LambertW(-x))*(1+x)).

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A102743 Expansion of e.g.f. LambertW(-x)/(x*(x-1)).

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A277507 E.g.f.: -1/((1-LambertW(-x))*(1-x)).

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A344229 a(n) = n*a(n-1) + n^signum(n mod 4), a(0) = 1.

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A300519 Convolution of n! and n^n.

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A356691 a(n) = n! * Sum_{k=0..n} k^(2*k)/k!.

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A360596 Expansion of e.g.f. 1/( (1 - x) * (1 + LambertW(-2*x)) ).

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A374844 a(n) = n! * Sum_{k=1..n} k^k / k!.