A277452 -id:A277452 - 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

A332408 a(n) = Sum_{k=0..n} binomial(n,k) * k! * k^n.

Original entry on oeis.org

1, 1, 10, 213, 8284, 513105, 46406286, 5772636373, 945492503320, 197253667623681, 51069324556151290, 16067283861476491941, 6037615013420387657844, 2670812587802323522405393, 1373842484756310928089102022, 813119045938378747809030359445

Offset: 0

Ilya Gutkovskiy, Apr 23 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..232

Cf. A000522, A006153, A031971, A063170, A072034, A256016, A277452, A332627.

Join[{1}, Table[Sum[Binomial[n, k] k! k^n, {k, 0, n}], {n, 1, 15}]]

a(n) = sum(k=0, n, binomial(n,k) * k! * k^n); \\ Michel Marcus, Apr 24 2020

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x*exp(x))^k))) \\ Seiichi Manyama, Feb 19 2022

G.f.: Sum_{k>=0} k! * k^k * x^k / (1 - k*x)^(k+1).
a(n) = n! * Sum_{k=0..n} k^n / (n-k)!.
a(n) ~ c * n! * n^n, where c = A073229 = exp(exp(-1)). - Vaclav Kotesovec, Feb 20 2021
E.g.f.: Sum_{k>=0} (k*x*exp(x))^k. - Seiichi Manyama, Feb 19 2022

A319392 a(n) = Sum_{k=0..n} (-1)^(n-k)binomial(n,k)k!*n^k.

Original entry on oeis.org

1, 0, 5, 116, 4785, 307024, 28435285, 3598112580, 596971515329, 125802906617600, 32834740225688901, 10399056510149276980, 3929349957207906673585, 1746371472945523953503376, 901944505258819679842017365, 535692457387043907059336566724, 362573376628272441934460817960705

Offset: 0

Ilya Gutkovskiy, Sep 18 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..232

Main diagonal of A320032.

Cf. A000166, A000180, A000354, A001907, A001908, A277452.

b:= proc(n, k) option remember;
     `if`(n=0, 1, k*n*b(n-1, k)+(-1)^n)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..17);  # Alois P. Heinz, May 07 2020

Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n, k] k! n^k, {k, 0, n}], {n, 16}]]
Table[n! SeriesCoefficient[Exp[-x]/(1 - n x), {x, 0, n}], {n, 0, 16}]
Table[(-1)^n HypergeometricPFQ[{1, -n}, {}, n], {n, 0, 16}]

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n,k)*k!*n^k); \\ Michel Marcus, Sep 18 2018

a(n) = n! * [x^n] exp(-x)/(1 - n*x).
a(n) = exp(-1/n)*n^n*Gamma(n+1,-1/n) for n > 0, where Gamma(a,x) is the incomplete gamma function.
a(n) ~ n! * n^n. - Vaclav Kotesovec, Jun 09 2019

A320031 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of the e.g.f. exp(x)/(1 - k*x).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 13, 16, 1, 1, 5, 25, 79, 65, 1, 1, 6, 41, 226, 633, 326, 1, 1, 7, 61, 493, 2713, 6331, 1957, 1, 1, 8, 85, 916, 7889, 40696, 75973, 13700, 1, 1, 9, 113, 1531, 18321, 157781, 732529, 1063623, 109601, 1, 1, 10, 145, 2374, 36745, 458026, 3786745, 15383110, 17017969, 986410, 1

Offset: 0

Ilya Gutkovskiy, Oct 03 2018

			E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (2*k^2 + 2*k + 1)*x^2/2! + (6*k^3 + 6*k^2 + 3*k + 1)*x^3/3! + (24*k^4 + 24*k^3 + 12*k^2 + 4*k + 1)*x^4/4! + ...
Square array begins:
  1,    1,     1,      1,       1,       1,  ...
  1,    2,     3,      4,       5,       6,  ...
  1,    5,    13,     25,      41,      61,  ...
  1,   16,    79,    226,     493,     916,  ...
  1,   65,   633,   2713,    7889,   18321,  ...
  1,  326,  6331,  40696,  157781,  458026,  ...

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

Columns k=0..6 give A000012, A000522, A010844, A010845, A056545, A056546, A056547.
Main diagonal gives A277452.
Cf. A007318, A320032, A371898.

A := (n, k) -> simplify(hypergeom([1, -n], [], -k)):
for n from 0 to 5 do seq(A(n, k), k=0..8) od; # Peter Luschny, Oct 03 2018
# second Maple program:
A:= proc(n, k) option remember;
      1 + `if`(n>0, k*n*A(n-1, k), 0)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, May 09 2020

Table[Function[k, n! SeriesCoefficient[Exp[x]/(1 - k x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, HypergeometricPFQ[{1, -n}, {}, -k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

E.g.f. of column k: exp(x)/(1 - k*x).
A(n,k) = Sum_{j=0..n} binomial(n,j)*j!*k^j.
A(n,k) = hypergeom_2F0([1, -n], [], -k).
A(n,k) = 1 + [n > 0] * k * n * A(n-1,k). - Alois P. Heinz, May 09 2020
A(n,k) = floor(n!*k^n*exp(1/k)), k > 0, n + k > 1. - Peter McNair, Dec 20 2021
From Werner Schulte, Apr 14 2024: (Start)
The LU decomposition of this array is given by the upper triangular matrix U which is the transpose of A007318 and the lower triangular matrix L = A371898, i.e., A(n, k) = Sum_{i=0..k} binomial(k, i) * A371898(n, i).
Conjecture: E.g.f. of row n is exp(x) * (Sum_{k=0..n} A371898(n, k) * x^k / k!). (End)

A330260 a(n) = n! * Sum_{k=0..n} binomial(n,k) * n^(n - k) / k!.

Original entry on oeis.org

1, 2, 17, 352, 13505, 830126, 74717857, 9263893892, 1513712421377, 315230799073690, 81499084718806001, 25612081645835777192, 9615370149488574778177, 4250194195208050117007942, 2184834047906975645398282625, 1292386053018890618812398220876

Offset: 0

Ilya Gutkovskiy, Dec 18 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..232

Cf. A002720, A025167, A061711, A102757, A102773, A187021, A277373, A277452, A293146, A330497.

Main diagonal of A341014.

[Factorial(n)*&+[Binomial(n,k)*n^(n-k)/Factorial(k):k in [0..n]]:n in [0..15]]; // Marius A. Burtea, Dec 18 2019

Join[{1}, Table[n! Sum[Binomial[n, k] n^(n - k)/k!, {k, 0, n}], {n, 1, 15}]]
Join[{1}, Table[n^n n! LaguerreL[n, -1/n], {n, 1, 15}]]
Table[n! SeriesCoefficient[Exp[x/(1 - n x)]/(1 - n x), {x, 0, n}], {n, 0, 15}]

a(n) = n! * sum(k=0, n, binomial(n,k) * n^(n-k)/k!); \\ Michel Marcus, Dec 18 2019

a(n) = n! * [x^n] exp(x/(1 - n*x)) / (1 - n*x).
a(n) = Sum_{k=0..n} binomial(n,k)^2 * n^k * k!.
a(n) ~ sqrt(2*Pi) * BesselI(0,2) * n^(2*n + 1/2) / exp(n). - Vaclav Kotesovec, Dec 18 2019

A277453 a(n) = Sum_{k=0..n} binomial(n,k) * 2^k * n^k * k!.

Original entry on oeis.org

1, 3, 41, 1531, 111393, 13262051, 2336744233, 570621092091, 184341785557121, 76092709735150723, 39064090158380196201, 24408768326642565035963, 18237590837527919131499041, 16056004231253610384348995811, 16448689708899063469247204152553

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

Cf. A086331, A277373, A277423, A277452.

Flatten[{1, Table[Sum[Binomial[n, k]*2^k*n^k*k!, {k, 0, n}], {n, 1, 20}]}]

a(n) = exp(1/(2*n)) * 2^n * n^n * Gamma(n+1, 1/(2*n)).

a(n) ~ 2^n * n^n * n!.

cp's OEIS Frontend

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

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A332408 a(n) = Sum_{k=0..n} binomial(n,k) * k! * k^n.

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A319392 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*k!*n^k.

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A320031 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of the e.g.f. exp(x)/(1 - k*x).

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A330260 a(n) = n! * Sum_{k=0..n} binomial(n,k) * n^(n - k) / k!.

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Magma

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A277453 a(n) = Sum_{k=0..n} binomial(n,k) * 2^k * n^k * k!.

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Mathematica

Formula

A319392 a(n) = Sum_{k=0..n} (-1)^(n-k)binomial(n,k)k!*n^k.