A256016 -id:A256016 - cp's OEIS frontend

A031971 a(n) = Sum_{k=1..n} k^n.

1, 5, 36, 354, 4425, 67171, 1200304, 24684612, 574304985, 14914341925, 427675990236, 13421957361110, 457593884876401, 16841089312342855, 665478473553144000, 28101527071305611528, 1262899292504270591313, 60182438244917445266889, 3031284048960901518840700

Offset: 1

Chris du Feu (chris(AT)beckingham0.demon.co.uk)

From Alexander Adamchuk, Jul 21 2006: (Start)
p^(3k - 1) divides a(p^k) for prime p > 2 and k = 1, 2, 3, 4, ... or p^2 divides a(p) for prime p > 2. p^5 divides a(p^2) for prime p > 2. p^8 divides a(p^3) for prime p > 2. p^11 divides a(p^4) for prime p > 2.
p^2 divides a(2p) for prime p > 3. p^3 divides a(3p) for prime p > 2. p^2 divides a(4p) for prime p > 5. p^3 divides a(5p) for prime p > 3. p^2 divides a(6p) for prime p > 7.
p divides a(2p - 1) for all prime p. p^3 divides a(2p^2 - 1) for all prime p. p^5 divides a(2p^3 - 1) for all prime p.
p divides a((p - 1)/2) for p = 5, 13, 17, 29, 37, 41, 53, 61, ... = A002144 Pythagorean primes: primes of form 4n + 1.
(End)
If p prime then a(p-1) == -1 (mod p) [see De Koninck & Mercier reference]. Example: for p = 7, a(6) = 67171 = 7 * 9596 - 1. - Bernard Schott, Mar 06 2020

J.-M. De Koninck et A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 327 pp. 48-200, Ellipses, Paris (2004).
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 21.

T. D. Noe, Table of n, a(n) for n = 1..100
J. M. Grau, A. M. Oller-Marcen, and J. Sondow, On the congruence 1^m + 2^m + ... + m^m == n (mod m) with n|m, arXiv 1309.7941 [math.NT], 2013-2014.

A diagonal of array A103438.
Cf. A002144, A014117, A089072, A229303, A256016, A261490.
For a(n) mod n see A182398.

a031971 = sum . a089072_row  -- Reinhard Zumkeller, Mar 18 2013

[&+[(k)^n: k in [0..n]]: n in [1..30]]; // Vincenzo Librandi, Apr 18 2011

a := n->sum('i'^n,'i'=1..n);
# alternative
A031971 := proc(n)
    (bernoulli(n+1,n+1)-bernoulli(n+1))/(n+1) ;
end proc: # R. J. Mathar, May 10 2013

Table[Zeta[-n] - Zeta[-n, n + 1], {n, 25}] (* Alexander Adamchuk, Jul 21 2006 *)
Table[Total[Range[n]^n], {n,25}] (* T. D. Noe, Apr 19 2011 *)
Table[HarmonicNumber[n, -n], {n, 1, 25}] (* Jean-François Alcover, Apr 09 2015 *)

a(n)=sum(k=1,n,k^n) \\ Charles R Greathouse IV, Jun 05 2015

from sympy import harmonic
def A031971(n):
    return harmonic(n,-n) # Chai Wah Wu, Feb 15 2020

a(n) is asymptotic to (e/(e - 1))*n^n. - Benoit Cloitre, Dec 17 2003
a(n) = zeta(-n) - zeta(-n, n + 1), where zeta(s) is the Riemann zeta function and zeta(s, a) is the Hurwitz zeta function, a generalization of the Riemann zeta function. - Alexander Adamchuk, Jul 21 2006
a(n) == 1 (mod n) <==> n is in A014117 = 1, 2, 6, 42, 1806 (see the link "On the congruence ..."). - Jonathan Sondow, Oct 18 2013
a(A054377(n)) = A233045(n). - Jonathan Sondow, Dec 11 2013
a(n) = n! * [x^n] exp(x)*(exp(n*x) - 1)/(exp(x) - 1). - Ilya Gutkovskiy, Apr 07 2018
a(n) ~ ((e*n+1)/((e-1)*(n+1))) * n^n. - N. J. A. Sloane, Oct 13 2018, based on email from Claude F. Leibovici who claims this is slightly better than Cloitre's version when n is small.

A072034 a(n) = Sum_{k=0..n} binomial(n,k)*k^n.

Original entry on oeis.org

1, 1, 6, 54, 680, 11000, 217392, 5076400, 136761984, 4175432064, 142469423360, 5372711277824, 221903307604992, 9961821300640768, 482982946946734080, 25150966159083264000, 1400031335107317628928, 82960293298087664648192

Offset: 0

Karol A. Penson, Jun 07 2002

nonn

The number of functions from {1,2,...,n} into a subset of {1,2,...,n} summed over all subsets. - Geoffrey Critzer, Sep 16 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
V. Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013

Cf. A088789, A242446, A256016, A336828, A341815.

seq(add(binomial(n,k)*k^n,k=0..n),n=0..17); # Peter Luschny, Jun 09 2015

Table[Sum[Binomial[n,k]k^n,{k,0,n}],{n,1,20}] (* Geoffrey Critzer, Sep 16 2012 *)

x='x+O('x^50); Vec(serlaplace(1/(1 + lambertw(-x*exp(x))))) \\ G. C. Greubel, Nov 10 2017

a(n) = sum(k=0, n, binomial(n,k)*k^n); \\ Michel Marcus, Nov 10 2017

E.g.f.: 1/(1+LambertW(-x*exp(x))). - Vladeta Jovovic, Mar 29 2008
a(n) ~ (n/(e*LambertW(1/e)))^n/sqrt(1+LambertW(1/e)). - Vaclav Kotesovec, Nov 26 2012
O.g.f.: Sum_{n>=0} n^n * x^n / (1 - n*x)^(n+1). - Paul D. Hanna, May 22 2018

Offset set to 0 and a(0) = 1 prepended by Peter Luschny, Jun 09 2015

E.g.f. edited to include a(0)=1 by Robert Israel, Jun 09 2015

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

A242446 a(n) = Sum_{k=1..n} C(n,k) * k^(2*n).

Original entry on oeis.org

1, 18, 924, 93320, 15609240, 3903974592, 1364509038592, 635177480713344, 379867490829555840, 283825251434680651520, 259092157573229145859584, 283735986144895532781391872, 367138254141051794797009309696, 554136240038549806366753446051840

Offset: 1

Vaclav Kotesovec, May 14 2014

Generally, for p>=1, a(n) = Sum_{k=1..n} C(n,k) * k^(p*n) is asymptotic to sqrt(r/(p+r-p*r)) * r^(p*n) * n^(p*n) / (exp(p*n) * (1-r)^n), where r = p/(p+LambertW(p*exp(-p))).

Sum_{k=1..n} (-1)^(n-k) * C(n,k) * k^(p*n) = n! * stirling2(p*n,n).

Cf. A007820, A072034, A242449, A256016.

Table[Sum[Binomial[n,k]*k^(2*n),{k,1,n}],{n,1,20}]

a(n) ~ sqrt(r/(2-r)) * r^(2*n) * n^(2*n) / (exp(2*n) * (1-r)^n), where r = 2/(2+LambertW(2*exp(-2))).

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

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

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

A356687 a(n) = n! * Sum_{k=0..n} k^(2*n)/k!.

Original entry on oeis.org

1, 1, 18, 927, 94876, 16251045, 4210190766, 1543550310211, 764096247603480, 493254380867214249, 404269328278061434810, 411862088865696890314311, 512690851568229926690616948, 768775988931240685277619894157

Offset: 0

Seiichi Manyama, Aug 23 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..196

Cf. A256016, A356688.

Cf. A030297, A249459, A356672.

a[n_] := n! * Sum[k^(2*n)/k!, {k, 0, n}]; a[0] = 1; Array[a, 14, 0] (* Amiram Eldar, Aug 23 2022 *)

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

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k^2*x)^k/(k!*(1-k^2*x)))))

E.g.f.: Sum_{k>=0} (k^2 * x)^k / (k! * (1 - k^2 * 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)).

A356689 a(n) = n! * Sum_{k=0..n} k^(k*n)/k!.

Original entry on oeis.org

1, 2, 20, 19887, 4297096180, 298028721722131825, 10314430386434427534836297166, 256923580889667624113335512704714686054849, 6277101737079381675512518990977258744796239498871290255000

Offset: 0

Seiichi Manyama, Aug 23 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..26

Cf. A256016, A356687, A356688.

Cf. A349886, A356674.

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

my(N=10, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k^k*x)^k/(k!*(1-k^k*x)))))

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

cp's OEIS Frontend

A031971 a(n) = Sum_{k=1..n} k^n.

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A072034 a(n) = Sum_{k=0..n} binomial(n,k)*k^n.

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

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A242446 a(n) = Sum_{k=1..n} C(n,k) * k^(2*n).

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

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

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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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