A337002 -id:A337002 - cp's OEIS frontend

A256016 a(n) = n! * Sum_{k=0..n} k^n/k!.

1, 1, 6, 57, 796, 15145, 374526, 11669665, 447595800, 20733553809, 1141067915290, 73552752257281, 5484203261135028, 467864288815609465, 45236104846954021014, 4915818294874879570305, 596044703812665607374256, 80118478395137652912476449, 11870487496575403846760198322

Offset: 0

Vaclav Kotesovec, Jun 01 2015

Seiichi Manyama, Table of n, a(n) for n = 0..283
Eric Weisstein's World of Mathematics, Bell Polynomial.
Wikipedia, Touchard polynomials

Cf. A000110, A031971, A072034, A242446, A337001, A337002, A354436.

Main diagonal of A337085.

Join[{1}, Table[n!*Sum[k^n/k!,{k,0,n}],{n,1,20}]]

a(n) = n!*sum(k=0, n, k^n/k!); \\ Michel Marcus, Aug 15 2020

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x)^k/(k!*(1-k*x))))) \\ Seiichi Manyama, Aug 23 2022

a(n) ~ e*Bell(n)*n!, for the Bell numbers see A000110.
a(n) ~ sqrt(2*Pi) * n^(2*n+1/2) * exp(n/LambertW(n)-2*n) / (sqrt(1+LambertW(n)) * LambertW(n)^n).
E.g.f.: Sum_{k>=0} (k * x)^k / (k! * (1 - k * x)). - Seiichi Manyama, Aug 23 2022
a(n) = n! * [x^n] B_n(x) * exp(x) / (1-x), where B_n(x) = Bell polynomials. - Seiichi Manyama, Jan 04 2024
a(n) = Sum_{k=0..n} k^n*(n-k)!*binomial(n,k). - Ridouane Oudra, Jun 16 2025

a(0)=1 prepended by Seiichi Manyama, Aug 14 2020

A337001 a(n) = n! * Sum_{k=0..n} k^3 / k!.

Original entry on oeis.org

0, 1, 10, 57, 292, 1585, 9726, 68425, 547912, 4931937, 49320370, 542525401, 6510306540, 84633987217, 1184875823782, 17773137360105, 284370197765776, 4834293362023105, 87017280516421722, 1653328329812019577, 33066566596240399540, 694397898521048399601

Offset: 0

Ilya Gutkovskiy, Aug 10 2020

nonn

Exponential convolution of cubes (A000578) and factorial numbers (A000142).

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A000142, A000522, A000578, A007526, A030297, A256016, A337002.

Row sums of A121682.

Table[n! Sum[k^3/k!, {k, 0, n}], {n, 0, 21}]
nmax = 21; CoefficientList[Series[x (1 + 3 x + x^2) Exp[x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = n (n^2 + a[n - 1]); Table[a[n], {n, 0, 21}]

a(n) = n! * sum(k=0, n, k^3/k!); \\ Michel Marcus, Aug 12 2020

E.g.f.: x * (1 + 3*x + x^2) * exp(x) / (1 - x).
a(0) = 0; a(n) = n * (n^2 + a(n-1)).
a(n) ~ 5*exp(1)*n!. - Vaclav Kotesovec, Jan 13 2024

A337085 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) = n! * Sum_{j=0..n} j^k/j!.

Original entry on oeis.org

1, 0, 2, 0, 1, 5, 0, 1, 4, 16, 0, 1, 6, 15, 65, 0, 1, 10, 27, 64, 326, 0, 1, 18, 57, 124, 325, 1957, 0, 1, 34, 135, 292, 645, 1956, 13700, 0, 1, 66, 345, 796, 1585, 3906, 13699, 109601, 0, 1, 130, 927, 2404, 4605, 9726, 27391, 109600, 986410, 0, 1, 258, 2577, 7804, 15145, 28926, 68425, 219192, 986409, 9864101

Offset: 0

Seiichi Manyama, Aug 14 2020

			Square array begins:
     1,    0,    0,    0,     0,     0,      0, ...
     2,    1,    1,    1,     1,     1,      1, ...
     5,    4,    6,   10,    18,    34,     66, ...
    16,   15,   27,   57,   135,   345,    927, ...
    65,   64,  124,  292,   796,  2404,   7804, ...
   326,  325,  645, 1585,  4605, 15145,  54645, ...
  1957, 1956, 3906, 9726, 28926, 98646, 374526, ...

Eric Weisstein's World of Mathematics, Bell Polynomial.
Wikipedia, Touchard polynomials

Columns k=0..5 give A000522, A007526, A030297, A337001, A337002, A368719.
Main diagonal gives A256016.
Cf. A368724.

T[n_, k_] := n! * Sum[If[j == k == 0, 1, j^k]/j!, {j, 0, n}]; Table[T[k, n-k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 29 2021 *)

T(0,k) = 0^k and T(n,k) = n^k + n * T(n-1,k) for n>0.

E.g.f. of column k: B_k(x) * exp(x) / (1-x), where B_n(x) = Bell polynomials. - Seiichi Manyama, Jan 04 2024

A368575 a(n) = n! * Sum_{k=0..n} binomial(k+3,4) / k!.

Original entry on oeis.org

0, 1, 7, 36, 179, 965, 5916, 41622, 333306, 3000249, 30003205, 330036256, 3960436437, 51485675501, 720799459394, 10811991893970, 172991870307396, 2940861795230577, 52935512314156371, 1005774733968978364, 20115494679379576135, 422425388266971109461

Offset: 0

Seiichi Manyama, Dec 31 2023

Cf. A007526, A103519, A368574, A368576.

Cf. A000332, A337002.

my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(x*sum(k=0, 3, binomial(3, k)*x^k/(k+1)!)*exp(x)/(1-x))))

a(0) = 0; a(n) = n*a(n-1) + binomial(n+3,4).

E.g.f.: x * (1+3*x/2+x^2/2+x^3/24) * exp(x) / (1-x).

A368717 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^4 / k!.

Original entry on oeis.org

0, 1, 14, 39, 100, 125, 546, -1421, 15464, -132615, 1336150, -14683009, 176216844, -2290790411, 32071104170, -481066511925, 7697064256336, -130850092274191, 2355301661040414, -44750731559637545, 895014631192910900, -18795307255050934419

Offset: 0

Seiichi Manyama, Jan 04 2024

sign

Robert Israel, Table of n, a(n) for n = 0..448
Eric Weisstein's World of Mathematics, Bell Polynomial.
Wikipedia, Touchard polynomials

Column k=4 of A368724.

Cf. A048993, A337002, A368586.

f:= proc(n) option remember;
  - n*procname(n-1)+n^4
end proc:
f(0):= 0:
seq(f(i),i=0..30); # Robert Israel, May 13 2025

Table[-n + 2*n^2 + n^3 + (-1)^n*n*Subfactorial[n-1], {n, 0, 20}] (* Vaclav Kotesovec, Jul 18 2025 *)

my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, 4, stirling(4, k, 2)*x^k)*exp(x)/(1+x))))

a(0) = 0; a(n) = -n*a(n-1) + n^4.
E.g.f.: B_4(x) * exp(x) / (1+x), where B_n(x) = Bell polynomials.
a(n) ~ (-1)^n * exp(-1) * n!. - Vaclav Kotesovec, Jul 18 2025

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

A256016 a(n) = n! * Sum_{k=0..n} k^n/k!.

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A337001 a(n) = n! * Sum_{k=0..n} k^3 / k!.

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A337085 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) = n! * Sum_{j=0..n} j^k/j!.

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A368575 a(n) = n! * Sum_{k=0..n} binomial(k+3,4) / k!.

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A368717 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^4 / k!.

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

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