A031971 -id:A031971 - cp's OEIS frontend

A261490 The total element sum of the n-fold f transform applied to the length n sequence of 1's. And f returns a sorted result after multiplying the elements in its input sequence with 1, 2, 3,... in descending size order.

0, 1, 4, 19, 100, 633, 4626, 37878, 348224, 3542952, 39339852, 478962252, 6289532928, 89038853856, 1346224983936, 21729308136720, 371924399416320, 6740200653419520, 128878557725067264, 2598800542616444724, 54986036469506668800, 1217069235297874269792

Offset: 0

Alois P. Heinz, Aug 21 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250

Main diagonal of A166278.

Cf. A031971 (for ascending sort), A036740 (when sum is replaced by product).

f:= l-> sort([seq(sort(l, `>`)[i]*i, i=1..nops(l))]):
a:= n-> add(i, i=(f@@n)([1$n])):
seq(a(n), n=0..35);

A344429 a(n) = Sum_{k=1..n} mu(k) * k^n.

Original entry on oeis.org

1, -3, -34, -96, -3399, 30239, -624046, -4482626, -32249230, 9768165230, -186975207617, -2150337557747, -327482869358214, 6894274639051756, 539094536846680025, 8044964790023844733, -707278869236116107432, -12275330572755863672628, -2190860499375418948848067

Offset: 1

Seiichi Manyama, May 19 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A002321, A008683, A031971, A068340, A321222, A332468, A336276, A336277, A336278, A336279, A344430.

a[n_] := Sum[MoebiusMu[k] * k^n, {k,1,n}]; Array[a, 20] (* Amiram Eldar, May 19 2021 *)

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

from functools import lru_cache
from math import comb
from sympy import bernoulli
@lru_cache(maxsize=None)
def faulhaber(n,p):
    """ Faulhaber's formula for calculating Sum_{k=1..n} k^p
        requires sympy version 1.12+ where bernoulli(1) = 1/2
    """
    return sum(comb(p+1,k)*bernoulli(k)*n**(p-k+1) for k in range(p+1))//(p+1)
@lru_cache(maxsize=None)
def A344429(n,m=None):
    if n <= 1:
        return 1
    if m is None:
        m=n
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (faulhaber(j-1,m)-faulhaber(j2-1,m))*A344429(k1,m)
        j, k1 = j2, n//j2
    return c+faulhaber(j-1,m)-faulhaber(n,m) # Chai Wah Wu, Nov 02 2023

A349836 Expansion of Sum_{k>=0} (k * x)^k/(1 - k^2 * x).

Original entry on oeis.org

1, 1, 5, 44, 564, 9665, 211025, 5686104, 184813048, 7118824417, 320295658577, 16626717667348, 985178854556524, 66005199079345025, 4958773228726876257, 414664315430994701616, 38344259607889223269168, 3898112616839310343827009

Offset: 0

Seiichi Manyama, Dec 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..299

Cf. A031971, A349883.

Cf. A249459, A349884, A368505.

Join[{1}, Table[Sum[k^(2*n - k), {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Dec 04 2021 *)

a(n, t=2) = sum(k=0, n, k^(t*(n-k)+k));

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

a(n) = Sum_{k=0..n} k^(2*n-k).

a(n) ~ sqrt(Pi) * 2^(1 + 2*n - 2*n/LambertW(2*exp(1)*n)) * (n/LambertW(2*exp(1)*n))^(1/2 + 2*n - 2*n/LambertW(2*exp(1)*n)) / sqrt(1 + LambertW(2*exp(1)*n)). - Vaclav Kotesovec, Dec 04 2021

A349883 Expansion of Sum_{k>=0} (k * x)^k/(1 - k^3 * x).

Original entry on oeis.org

1, 1, 5, 60, 1242, 41241, 2033683, 141318208, 13262986788, 1624337451945, 252725477615989, 48858277079478156, 11523986801592238046, 3265676705193282018577, 1097336766468309067029991, 432291795385094609190468384, 197690320046319097006619353352

Offset: 0

Seiichi Manyama, Dec 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..227

Cf. A031971, A349836, A349901.

a[n_] := Sum[If[k == 3*n - 2*k == 0, 1, k^(3*n - 2*k)], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Dec 04 2021 *)

a(n, t=3) = sum(k=0, n, k^(t*(n-k)+k));

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

a(n) = Sum_{k=0..n} k^(3*n-2*k).

a(n) ~ sqrt(Pi) * (3/2)^(1/2 + 3*n - 3*n/LambertW(3*exp(1)*n/2)) * (n/LambertW(3*exp(1)*n/2))^(1/2 + 3*n - 3*n/LambertW(3*exp(1)*n/2)) / sqrt(1 + LambertW(3*exp(1)*n/2)). - Vaclav Kotesovec, Dec 04 2021

A349963 a(n) = Sum_{k=0..n} (2*k)^n.

Original entry on oeis.org

1, 2, 20, 288, 5664, 141600, 4298944, 153638912, 6319260672, 294044152320, 15272286131200, 875880428003328, 54976337351106560, 3748609104907476992, 275924407293425336320, 21806398621389422592000, 1841661678145084557099008, 165530736067119754944577536

Offset: 0

Seiichi Manyama, Dec 07 2021

nonn

Cf. A031971, A185393, A249459, A349970.

a[n_] := Sum[If[k == n == 0, 1, (2*k)^n], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Dec 07 2021 *)

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

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

G.f.: Sum_{k>=0} (2*k * x)^k/(1 - 2*k * x).
a(n) = 2^n * A031971(n).
a(n) ~ c * 2^n * n^n, where c = 1/(1 - 1/exp(1)) = A185393. - Vaclav Kotesovec, Dec 07 2021

A353013 a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^(n-k).

Original entry on oeis.org

1, 1, 4, 28, 264, 3207, 47696, 839412, 17061280, 393264145, 10135913792, 288839201432, 9017184333440, 306045200463519, 11220008681600256, 441866073895351128, 18603606156815235584, 833860238440653331505, 39643749441387211150336

Offset: 0

Seiichi Manyama, Apr 16 2022

Cf. A031971, A353014.

Cf. A352981, A352944.

a[0] = 1; a[n_] := Sum[(n - 2*k)^(n - k), {k, 0, Floor[n/2]}]; Array[a, 20, 0] (* Amiram Eldar, Apr 16 2022 *)

PARI
```
a(n) = sum(k=0, n\2, (n-2*k)^(n-k));
```

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

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

A353014 a(n) = Sum_{k=0..floor(n/3)} (n-3k)^(n-2k).

Original entry on oeis.org

1, 1, 4, 27, 257, 3133, 46737, 824568, 16792857, 387700668, 10005768898, 285445966496, 8919588913002, 302975146962245, 11115146328067250, 438000914977377939, 18450682450377791691, 827395864513198608177, 39352977767853205024131

Offset: 0

Seiichi Manyama, Apr 16 2022

Cf. A031971, A353013.

Cf. A352946, A353015.

a[0] = 1; a[n_] := Sum[(n - 3*k)^(n - 2*k), {k, 0, Floor[n/3]}]; Array[a, 20, 0] (* Amiram Eldar, Apr 16 2022 *)

PARI
```
a(n) = sum(k=0, n\3, (n-3*k)^(n-2*k));
```

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

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

A074209 a(n) = Sum_{i=n+1..2n} i^n.

Original entry on oeis.org

2, 25, 405, 8418, 216400, 6668779, 240361121, 9936764996, 463893277176, 24148657338925, 1387253043076813, 87185783860333910, 5951020164442347800, 438417132703015536399, 34673851743509883542625

Offset: 1

Zak Seidov, Sep 22 2002

nonn

A rapidly growing sequence. An even more rapidly growing sequence, the sum of next n terms of the form i^i, is given in A074309. Sum of first n terms of the form i^n is A031971. Sum of first n terms of the form i^i is A001923.

			a(2) = 25 = 3^2 + 4^2, a(3) = 405 = 4^3 + 5^3 + 6^3, a(4) = 8418 = 5^4 + 6^4 + 7^4 + 8^4, a(5) = 216400 = 6^4 + 7^5 + 8^5 + 9^5 + 10^5.

Seiichi Manyama, Table of n, a(n) for n = 1..351

Cf. A001923, A031971, A074309.

Mathematica
```
Table[Sum[i^n, {i, n+1, 2n}], {n, 20}]
```

a(n) = sum(k=n+1, 2*n, k^n); \\ Seiichi Manyama, Dec 05 2021

From Wesley Ivan Hurt, Jan 28 2021: (Start)
a(n) = Sum_{k=1..n} (n+k)^n.
a(n) = Zeta(-n,n+1) - Zeta(-n,2*n+1), where Zeta is the Hurwitz zeta function. (End)
a(n) ~ (2*n)^n / (1 - exp(-1/2)). - Vaclav Kotesovec, Dec 06 2021

Name changed by Wesley Ivan Hurt, Jan 28 2021

A182398 a(n) = (Sum_{k=1..2n} k^2n) mod 2n.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 7, 8, 3, 6, 11, 4, 13, 14, 5, 16, 17, 6, 19, 12, 1, 22, 23, 8, 25, 26, 9, 28, 29, 58, 31, 32, 11, 34, 35, 12, 37, 38, 13, 24, 41, 2, 43, 44, 15, 46, 47, 16, 49, 30, 17, 52, 53, 18, 45, 56, 19, 58, 59, 116, 61, 62, 3, 64, 65, 22, 67, 68, 23

Offset: 1

Michel Lagneau, Apr 27 2012

nonn

Sum_{k=1..n} k^n (mod n) = 0 if n odd.
Properties of this sequence:
a(n) = 1 for n = 1, 3, 21, 903, ...
a(n) = n if n not divisible by 3;
a(3*n) = n except for n = 7, 10, 14, 20, 21, 26, 28, 30, 35, ...
a(21*n) = n, except for n = 10, 20, 26, 30, 40, 43, 50, 52, ...
a(903*n) = n, except for n = 10, ....
It appears that a(A007018(n)/2) = 1 and conjecturally a(m*A007018(n)/2) = m for a majority of value m.
No, a(A007018(n)/2) <> 1 for n > 4. (For example, a(A007018(5)/2) = a(1631721) = 1807.) - Jonathan Sondow, Oct 18 2013
0 < a(n) < 10 for n: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 18, 21, 24, 27, 42, 63, 84, 105, 126, 147, 168, 189, 903, 1806, 2709, 3612, 4515, 5418, 6321, 7224, 8127, .... Search limit was 25000. - Robert G. Wilson v, Jun 18 2015

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms 1..2499 from Michel Lagneau)
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, 2013.
Mathematics Stack Exchange, 1^n + 2^n + ... + (p-1)^n mod p = ?

Cf. A007018, A014117, A031971, A229307, A229311.

for n from 1 to 100 do: s:=sum('k^(2*n)', 'k'=1..2*n)
: x:=irem(s,2*n): printf(`%d, `,x):od:
# second Maple program:
a:= n-> add(k&^(2*n) mod (2*n), k=1..2*n) mod (2*n):
seq(a(n), n=1..100);

Table[Mod[Total[PowerMod[Range[2*n], 2*n, 2*n]], 2*n], {n, 100}] (* T. D. Noe, Apr 28 2012 *)

a(n) = A031971(2n) mod 2n. - Jonathan Sondow, Oct 18 2013

A215078 Triangle of sums of the first k n-th powers multiplied by binomial(n,k), read by rows.

Original entry on oeis.org

0, 0, 1, 0, 2, 5, 0, 3, 27, 36, 0, 4, 102, 392, 354, 0, 5, 330, 2760, 6500, 4425, 0, 6, 975, 15880, 73350, 123090, 67171, 0, 7, 2709, 81060, 654500, 2033325, 2637327, 1200304, 0, 8, 7196, 381808, 5064780, 25926824, 59992660, 63259168, 24684612, 0, 9, 18468, 1696464, 35574840, 281668590, 1034305524, 1896003648, 1681960464, 574304985, 0, 10, 46125, 7208880, 232816500, 2740317300, 14981494710, 42457884000, 64240088580, 49143419250, 14914341925

Offset: 0

Olivier Gérard, Aug 02 2012

If one starts the sum at j=0, the initial term T(0,0) is 1.

			  0
  0  1
  0  2    5
  0  3   27       36
  0  4  102      392      354
  0  5  330     2760     6500     4425
  0  6  975    15880    73350   123090    67171
  0  7 2709    81060   654500  2033325  2637327  1200304

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Binomial convolution of A215083.

Cf. A215077 (row sums), A031971 (diagonal)

A215078 := proc(n,k)
        binomial(n,k)*add(j^n,j=1..k) ;
end proc:
seq(seq(A215078(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Jan 27 2023

Flatten[Table[Table[Sum[j^n, {j, 1, k}]*Binomial[n, k], {k, 0, n}], {n, 0, 10}], 1]

T(n,k) = binomial(n,k)*sum(j^n, j=1..k)

cp's OEIS Frontend

A261490 The total element sum of the n-fold f transform applied to the length n sequence of 1's. And f returns a sorted result after multiplying the elements in its input sequence with 1, 2, 3,... in descending size order.

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A344429 a(n) = Sum_{k=1..n} mu(k) * k^n.

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Python

A349836 Expansion of Sum_{k>=0} (k * x)^k/(1 - k^2 * x).

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A349883 Expansion of Sum_{k>=0} (k * x)^k/(1 - k^3 * x).

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Mathematica

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A349963 a(n) = Sum_{k=0..n} (2*k)^n.

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PARI

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A353013 a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^(n-k).

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PARI

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A353014 a(n) = Sum_{k=0..floor(n/3)} (n-3*k)^(n-2*k).

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Mathematica

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PARI

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A074209 a(n) = Sum_{i=n+1..2n} i^n.

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A353014 a(n) = Sum_{k=0..floor(n/3)} (n-3k)^(n-2k).