A337001 -id:A337001 - 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

A337002 a(n) = n! * Sum_{k=0..n} k^4 / k!.

Original entry on oeis.org

0, 1, 18, 135, 796, 4605, 28926, 204883, 1643160, 14795001, 147960010, 1627574751, 19530917748, 253901959285, 3554627468406, 53319412076715, 853110593292976, 14502880086064113, 261051841549259010, 4959984989436051511, 99199699788721190220

Offset: 0

Ilya Gutkovskiy, Aug 10 2020

nonn

Exponential convolution of fourth powers (A000583) and factorial numbers (A000142).

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

Cf. A000142, A000522, A000583, A007526, A030297, A256016, A337001.

Table[n! Sum[k^4/k!, {k, 0, n}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[x (1 + 7 x + 6 x^2 + x^3) Exp[x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = n (n^3 + a[n - 1]); Table[a[n], {n, 0, 20}]

a(n) = n! * sum(k=0, n, k^4/k!); \\ Michel Marcus, Aug 12 2020

E.g.f.: x * (1 + 7*x + 6*x^2 + x^3) * exp(x) / (1 - x).
a(0) = 0; a(n) = n * (n^3 + a(n-1)).
a(n) ~ 15*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

A368574 a(n) = n! * Sum_{k=0..n} binomial(k+2,3) / k!.

Original entry on oeis.org

0, 1, 6, 28, 132, 695, 4226, 29666, 237448, 2137197, 21372190, 235094376, 2821132876, 36674727843, 513446190362, 7701692856110, 123227085698576, 2094860456876761, 37707488223782838, 716442276251875252, 14328845525037506580, 300905756025787639951, 6619926632567328080946

Offset: 0

Seiichi Manyama, Dec 31 2023

Cf. A007526, A103519, A368575, A368576.

Cf. A000292, A337001.

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

a(0) = 0; a(n) = n*a(n-1) + binomial(n+2,3).

E.g.f.: x * (1+x+x^2/6) * exp(x) / (1-x).

A356688 a(n) = n! * Sum_{k=0..n} k^(3*n)/k!.

Original entry on oeis.org

1, 1, 66, 21225, 18952156, 36175231585, 126556309395486, 733064060959310689, 6540867625730306094360, 85180334386943946887707617, 1552697061493449955344530003290, 38315904135534199560725372265381721, 1245605749857294018587318829355458646068

Offset: 0

Seiichi Manyama, Aug 23 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..149

Cf. A256016, A356687.

Cf. A337001, A349901, A356673.

a[n_] := n! * Sum[k^(3*n)/k!, {k, 0, n}]; a[0] = 1; Array[a, 13, 0] (* Amiram Eldar, Aug 23 2022 *)

PARI
```
a(n) = n!*sum(k=0, n, k^(3*n)/k!);
```

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k^3*x)^k/(k!*(1-k^3*x)))))

E.g.f.: Sum_{k>=0} (k^3 * x)^k / (k! * (1 - k^3 * x)).

A121682 Triangle read by rows: T(i,j) = (T(i-1,j) + i)*i.

Original entry on oeis.org

1, 6, 4, 27, 21, 9, 124, 100, 52, 16, 645, 525, 285, 105, 25, 3906, 3186, 1746, 666, 186, 36, 27391, 22351, 12271, 4711, 1351, 301, 49, 219192, 178872, 98232, 37752, 10872, 2472, 456, 64, 1972809, 1609929, 884169, 339849, 97929, 22329, 4185, 657, 81, 19728190, 16099390, 8841790, 3398590, 979390, 223390, 41950, 6670, 910, 100

Offset: 1

Thomas Wieder, Aug 15 2006

The first column is A030297 = a(n) = n*(n+a(n-1)). The main diagonal are the squares A000290 = n^2. The first lower diagonal (6,21,52,...) is A069778 = q-factorial numbers 3!_q. See also A121662.

			Triangle begins:
      1
      6     4
     27    21     9
    124   100    52   16
    645   525   285  105  25
   3906  3186  1746  666  186  36
  27391 22351 12271 4711 1351 301 49
  ...

T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.

Cf. A030297, A000290, A069778, A121662.

Row sums give A337001.

T:= proc(i, j) option remember;
      `if`(j<1 or j>i, 0, (T(i-1, j)+i)*i)
    end:
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Jun 22 2022

T[n_, k_] /; 1 <= k <= n := T[n, k] = (T[n-1, k]+n)*n;
T[, ] = 0;
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 17 2022 *)

def T(i, j): return (T(i-1, j)+i)*i if 1 <= j <= i else 0
print([T(r, c) for r in range(1, 11) for c in range(1, r+1)]) # Michael S. Branicky, Jun 22 2022

Edited by N. J. A. Sloane, Sep 15 2006

Formula in name corrected by Alois P. Heinz, Jun 22 2022

A368716 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^3 / k!.

Original entry on oeis.org

0, 1, 6, 9, 28, -15, 306, -1799, 14904, -133407, 1335070, -14684439, 176214996, -2290792751, 32071101258, -481066515495, 7697064252016, -130850092279359, 2355301661034294, -44750731559644727, 895014631192902540, -18795307255050944079

Offset: 0

Seiichi Manyama, Jan 04 2024

sign

abs(a(n))/n is prime for n = 2, 3, 4, 5, 7, 13, 19, 28, 643 and no others up to n = 2000. - Robert Israel, May 13 2025

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

Column k=3 of A368724.

Cf. A048993, A337001, A368585.

f:= proc(n) option remember;
  - n*procname(n-1)+n^3
end proc:
f(0):= 0:
seq(f(i),i=0..30); # Robert Israel, May 13 2025

Table[n + n^2 + (-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, 3, stirling(3, k, 2)*x^k)*exp(x)/(1+x))))

a(0) = 0; a(n) = -n*a(n-1) + n^3.
E.g.f.: B_3(x) * exp(x) / (1+x), where B_n(x) = Bell polynomials.
a(n) ~ (-1)^n * exp(-1) * n!. - Vaclav Kotesovec, Jul 18 2025

A368760 a(n) = n! * (1 + Sum_{k=0..n} k^3 / k!).

Original entry on oeis.org

1, 2, 12, 63, 316, 1705, 10446, 73465, 588232, 5294817, 52949170, 582442201, 6989308140, 90861008017, 1272054114982, 19080811728105, 305292987653776, 5189980790119105, 93419654222149722, 1774973430220851577, 35499468604417039540, 745488840692757839601

Offset: 0

Seiichi Manyama, Jan 04 2024

nonn

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

Column k=3 of A368759.

Cf. A337001.

my(N=30, x='x+O('x^N)); Vec(serlaplace((1+sum(k=0, 3, stirling(3, k, 2)*x^k)*exp(x))/(1-x)))

a(0) = 1; a(n) = n*a(n-1) + n^3.
a(n) = n! + A337001(n).
E.g.f.: (1 + B_3(x) * exp(x)) / (1-x), where B_n(x) = Bell polynomials.

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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A337002 a(n) = n! * Sum_{k=0..n} k^4 / 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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A368574 a(n) = n! * Sum_{k=0..n} binomial(k+2,3) / k!.

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Formula

A356688 a(n) = n! * Sum_{k=0..n} k^(3*n)/k!.

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A121682 Triangle read by rows: T(i,j) = (T(i-1,j) + i)*i.

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A368716 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^3 / k!.

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A368760 a(n) = n! * (1 + Sum_{k=0..n} k^3 / k!).

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