A026898 -id:A026898 - cp's OEIS frontend

A341437 Numbers k such that k divides Sum_{j=0..k} j^(k-j).

1, 2, 6, 7, 9, 42, 46, 431, 1806, 2506, 11318, 16965, 25426, 33146, 33361, 37053, 49365, 99221, 224506, 359703, 436994

Offset: 1

Seiichi Manyama, Feb 11 2021

Numbers k such that k divides A026898(k-1).

a(19) > 10^5.

Do[If[Mod[Sum[PowerMod[k, n - k, n], {k, 0, n}], n] == 0, Print[n]], {n, 1, 3000}] (* Vaclav Kotesovec, Feb 12 2021 *)

isok(n) = sum(k=0, n, Mod(k, n)^(n-k))==0;

0^6 + 1^5 + 2^4 + 3^3 + 4^2 + 5^1 + 6^0 = 66 = 6 * 11. So 6 is a term.

a(19) from Vaclav Kotesovec, Feb 14 2021

a(20)-a(21) from Chai Wah Wu, Feb 15 2021

A344433 a(n) = Sum_{k=1..n} mu(k) * k^(n - k).

Original entry on oeis.org

1, 0, -2, -6, -17, -46, -132, -402, -1314, -4613, -17313, -68893, -288556, -1269637, -5907157, -29489299, -160431708, -955478664, -6145884133, -41584238971, -287650358748, -1984825313901, -13377544470631, -86142095523089, -512881404732949, -2634567148684612, -9205461936290915, 17544751152746927

Offset: 1

Seiichi Manyama, May 19 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..598

Cf. A026898, A292524, A344432.

a[n_] := Sum[MoebiusMu[k] * k^(n-k), {k,1,n}]; Array[a, 30] (* Amiram Eldar, May 19 2021 *)

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

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k/(1-k*x)))

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

A349881 Expansion of Sum_{k>=0} x^k/(1 - k^4 * x).

Original entry on oeis.org

1, 1, 2, 18, 339, 10915, 663140, 61264436, 8044351557, 1536980041573, 402558463751974, 137204787854813174, 60668198155262809815, 34351266752678243067591, 24185207999807747975188552, 20842786946335533698574605528

Offset: 0

Seiichi Manyama, Dec 03 2021

nonn

In general, for t>=1 and s>=0, Sum_{k=0..n} k^(t*(n-k)+s) ~ sqrt(2*Pi) * ((n + s/t)/LambertW(exp(1)*(n + s/t)))^(1/2 + (t*n + s) * (1 - 1/LambertW(exp(1)*(n + s/t)))) / sqrt(t*(1 + LambertW(exp(1)*(n + s/t)))). - Vaclav Kotesovec, Dec 04 2021

Seiichi Manyama, Table of n, a(n) for n = 0..195

Cf. A026898, A234568, A349880.

Cf. A349858.

a[n_] := Sum[If[k == n - k == 0, 1, k^(4*(n - k))], {k, 0, n}]; Array[a, 16, 0] (* Amiram Eldar, Dec 04 2021 *)

a(n, s=0, t=4) = sum(k=0, n, k^(t*(n-k)+s));

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k^4*x)))

a(n) = Sum_{k=0..n} k^(4*(n-k)).

a(n) ~ sqrt(Pi/2) * (n/LambertW(exp(1)*n))^(1/2 + 4*n - 4*n/LambertW(exp(1)*n)) / sqrt(1 + LambertW(exp(1)*n)). - Vaclav Kotesovec, Dec 04 2021

A349965 a(n) = Sum_{k=0..n} (k * (n-k))^k.

Original entry on oeis.org

1, 1, 2, 7, 47, 513, 8020, 169227, 4637965, 159568981, 6684686230, 332681461871, 19316990453131, 1292074091000105, 98636639620170792, 8528989125071254867, 829516920337723299465, 90124512307642049807293, 10865612430780251465538154

Offset: 0

Seiichi Manyama, Dec 07 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..286

Cf. A026898, A100262, A349966.

a[n_] := Sum[If[k == 0, 1, (k*(n - k))^k], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Dec 07 2021 *)

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

log(a(n)) ~ n*(2*log(n) - 1 + (1/(2*log(n)) - 1)*log(2*log(n))). - Vaclav Kotesovec, Dec 07 2021

A352946 a(n) = Sum_{k=0..floor(n/3)} (n-3*k)^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 6, 10, 16, 25, 42, 73, 125, 217, 391, 714, 1305, 2428, 4612, 8830, 17038, 33377, 66216, 132349, 267075, 545329, 1123693, 2333278, 4889751, 10342468, 22043954, 47340802, 102504532, 223654713, 491393646, 1087353601, 2423448817, 5437568233

Offset: 0

Seiichi Manyama, Apr 09 2022

Cf. A026898, A352944.

Cf. A001840.

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

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k*x^3)))

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

a(n) ~ sqrt(2*Pi/3) * (n/LambertW(exp(1)*n))^(n*(1 - 1/LambertW(exp(1)*n))/3 + 1/2) / sqrt(1 + LambertW(exp(1)*n)). - Vaclav Kotesovec, Apr 14 2022

A179928 Row sums of A179927, the triangle of centered orthotopic numbers.

Original entry on oeis.org

1, 3, 6, 13, 32, 89, 276, 943, 3514, 14159, 61242, 282633, 1384684, 7170701, 39105992, 223867419, 1341434134, 8392364851, 54696456734

Offset: 0

Peter Luschny, Aug 02 2010

nonn

a(n)-1 is the sum of the antidiagonal of array A265583 from (n+1,1) to (1,n+1). - Mathew Englander, Apr 11 2021

Cf. A179927, A265583, A026898.

A179928 := proc(n) local j; add(A179927(n,j),j=0..n) end;

e[0, ] = 1; e[n, x_] := e[n, x] = x (1 - x) D[e[n - 1, x], x] + e[n - 1, x] (1 + (n - 1) x) // Expand;
h[n_, x_] := e[n, x] (1 + x)/(1 - x)^(n + 1);
T[n_, k_] := SeriesCoefficient[h[n - k, x], {x, 0, k}];
a[n_] := Sum[T[n, k], {k, 0, n}];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jul 11 2019 *)

From Mathew Englander, Apr 11 2021: (Start)
a(n) = 1 + Sum_{i = 1..n} (i+1)*i^(n-i).
a(n) = A026898(n) + A026898(n-1) for n > 0. (End)

A242431 Triangle read by rows: T(n, k) = (k + 1)*T(n-1, k) + Sum_{j=k..n-1} T(n-1, j) for k < n, T(n, n) = 1. T(n, k) for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 14, 10, 4, 1, 43, 35, 17, 5, 1, 144, 128, 74, 26, 6, 1, 523, 491, 329, 137, 37, 7, 1, 2048, 1984, 1498, 730, 230, 50, 8, 1, 8597, 8469, 7011, 3939, 1439, 359, 65, 9, 1, 38486, 38230, 33856, 21568, 9068, 2588, 530, 82, 10, 1

Offset: 0

Peter Luschny, May 14 2014

			0|    1;
1|    2,    1;
2|    5,    3,    1;
3|   14,   10,    4,   1;
4|   43,   35,   17,   5,   1;
5|  144,  128,   74,  26,   6,  1;
6|  523,  491,  329, 137,  37,  7, 1;
7| 2048, 1984, 1498, 730, 230, 50, 8, 1;

Peter Luschny, Rows n = 0..50, flattened.
Mathew Englander, Comments on A101494 and A089246, and related sequences

Cf. A003101, A026898, A047969, A047970, A101494, A089246.

T := proc(n, k) option remember; local j;
    if k=n then 1
  elif k>n then 0
  else (k+1)*T(n-1, k) + add(T(n-1, j), j=k..n)
    fi end:
seq(print(seq(T(n,k), k=0..n)), n=0..7);

def A242431_rows():
    T = []; n = 0
    while True:
        T.append(1)
        yield T
        for k in (0..n):
            T[k] = (k+1)*T[k] + add(T[j] for j in (k..n))
        n += 1
a = A242431_rows()
for n in range(8): next(a)

T(n, 0) = A047970(n).
Sum_{k=0..n} T(n, k) = A112532(n+1).
From Mathew Englander, Feb 25 2021: (Start)
T(n,k) = 1 + Sum_{i = k+1..n} i*(i+1)^(n-i).
T(n,k) = T(n,k+1) + (k+1)*(k+2)^(n-k-1) for 0 <= k < n.
T(n,k) = T(n,k+1) + (k+2)*(T(n-1,k) - T(n-1,k+1)) for 0 <= k <= n-2.
T(n,k) = Sum_{i = 0..n-k} (k+2)^i*A089246(n-k,i).
Sum_{i = k..n} T(i,k) = Sum_{i = 0..n-k} (n+2-i)^i = Sum_{i = 0..n-k} A101494(n-k,i)*(k+2)^i. (End)

A349878 Expansion of Sum_{k>=0} k^3 * x^k/(1 - k * x).

Original entry on oeis.org

0, 1, 9, 44, 178, 689, 2723, 11304, 49772, 232657, 1151781, 6018628, 33087022, 190780001, 1150653679, 7241710656, 47454745496, 323154695841, 2282779990113, 16700904488284, 126356632389834, 987303454928465, 7957133905608283, 66071772829246808

Offset: 0

Seiichi Manyama, Dec 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A026898, A003101, A062809, A349879.

Cf. A349854.

Table[Sum[k^(n - k + 3), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 04 2021 *)

a(n, s=3, t=1) = sum(k=0, n, k^(t*(n-k)+s));

my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k^3*x^k/(1-k*x))))

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

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

A349879 Expansion of Sum_{k>=0} k^4 * x^k/(1 - k * x).

Original entry on oeis.org

0, 1, 17, 114, 564, 2507, 10961, 49260, 231928, 1150781, 6017297, 33085294, 190777804, 1150650935, 7241707281, 47454741400, 323154690928, 2282779984281, 16700904481425, 126356632381834, 987303454919204, 7957133905597635, 66071772829234641

Offset: 0

Seiichi Manyama, Dec 03 2021

nonn

In general, for s>=1, Sum_{k=0..n} k^(n-k+s) ~ sqrt(2*Pi) * ((n + s)/LambertW(exp(1)*(n + s)))^(1/2 + (n + s)*(1 - 1/LambertW(exp(1)*(n + s)))) / sqrt(1 + LambertW(exp(1)*(n + s))). - Vaclav Kotesovec, Dec 04 2021

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A026898, A003101, A062809, A349878.

Cf. A349855.

Table[Sum[k^(n - k + 4), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 04 2021 *)

a(n, s=4, t=1) = sum(k=0, n, k^(t*(n-k)+s));

my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k^4*x^k/(1-k*x))))

a(n) = Sum_{k=0..n} k^(n-k+4).

a(n) ~ sqrt(2*Pi) * ((n + 4)/LambertW(exp(1)*(n + 4)))^(1/2 + (n + 4)*(1 - 1/LambertW(exp(1)*(n + 4)))) / sqrt(1 + LambertW(exp(1)*(n + 4))). - Vaclav Kotesovec, Dec 04 2021

A349969 a(n) = Sum_{k=0..n} (k*n)^(n-k).

Original entry on oeis.org

1, 1, 3, 16, 141, 1871, 34951, 873174, 27951929, 1107415549, 52891809491, 2987861887924, 196828568831365, 14950745148070499, 1296606974501951743, 127238563043551898986, 14012626653816435643633, 1719136634276882827095009, 233448782800118609096218891

Offset: 0

Seiichi Manyama, Dec 07 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..284

Cf. A026898, A155956, A349964, A349970.

a[n_] := Sum[If[k == n == 0, 1, (k*n)^(n - k)], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Dec 07 2021 *)

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

a(n) = [x^n] Sum_{k>=0} x^k/(1 - n*k * x).

a(n) ~ sqrt(2*Pi/(n*(1 + LambertW(exp(1)*n^2)))) * (n^2/LambertW(exp(1)*n^2))^(n + 1/2 - n/LambertW(exp(1)*n^2)). - Vaclav Kotesovec, Dec 07 2021

cp's OEIS Frontend

A341437 Numbers k such that k divides Sum_{j=0..k} j^(k-j).

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A344433 a(n) = Sum_{k=1..n} mu(k) * k^(n - k).

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A349881 Expansion of Sum_{k>=0} x^k/(1 - k^4 * x).

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A349965 a(n) = Sum_{k=0..n} (k * (n-k))^k.

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A352946 a(n) = Sum_{k=0..floor(n/3)} (n-3*k)^k.

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A179928 Row sums of A179927, the triangle of centered orthotopic numbers.

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A242431 Triangle read by rows: T(n, k) = (k + 1)*T(n-1, k) + Sum_{j=k..n-1} T(n-1, j) for k < n, T(n, n) = 1. T(n, k) for n >= 0 and 0 <= k <= n.

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A349878 Expansion of Sum_{k>=0} k^3 * x^k/(1 - k * x).

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