A003101 -id:A003101 - cp's OEIS frontend

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

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

A349882 Expansion of Sum_{k>=0} k^2 * x^k/(1 - k^2 * x).

Original entry on oeis.org

0, 1, 5, 26, 162, 1267, 12343, 145652, 2036148, 33192789, 622384729, 13263528350, 318121600694, 8517247764135, 252725694989611, 8258153081400856, 295515712276222952, 11523986940937975401, 487562536078882116717, 22291094729329088403298

Offset: 0

Seiichi Manyama, Dec 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..339

Cf. A003101, A234568, A249459, A349863, A359659.

a[n_] := Sum[If[k == n - k + 1 == 0, 1, k^(2*(n - k + 1))], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Dec 04 2021 *)

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

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

my(N=20, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^k/(1-(k+1)^2*x)))) \\ Seiichi Manyama, Jan 12 2023

a(n) = Sum_{k=0..n} k^(2*(n-k+1)).
a(n) = A234568(n+1) - 1. - Hugo Pfoertner, Dec 04 2021
a(n) ~ sqrt(Pi) * ((n+1)/LambertW(exp(1)*(n+1)))^(5/2 + 2*n - 2*(n+1)/LambertW(exp(1)*(n+1))) / sqrt(1 + LambertW(exp(1)*(n+1))). - Vaclav Kotesovec, Dec 04 2021
G.f.: Sum_{k>=1} x^k/(1 - (k+1)^2 * x). - Seiichi Manyama, Jan 12 2023

A351279 a(n) = Sum_{k=0..n} 2^k * k^(n-k).

Original entry on oeis.org

1, 2, 6, 18, 58, 202, 762, 3114, 13754, 65386, 332922, 1806506, 10398266, 63226858, 404640250, 2716838186, 19083233210, 139874994282, 1067462826874, 8464760754602, 69620304280890, 592925117961450, 5220996124450042, 47467755352580650, 445027186867923642

Offset: 0

Seiichi Manyama, Feb 06 2022

Seiichi Manyama, Table of n, a(n) for n = 0..582

Cf. A003101, A026898, A038125, A349962.

a[0] = 1; a[n_] := Sum[2^k * k^(n-k), {k, 1, n}]; Array[a, 25, 0] (* Amiram Eldar, Feb 06 2022 *)

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

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (2*x)^k/(1-k*x)))

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

a(n) ~ sqrt(2*Pi/(1 + LambertW(exp(1)*n/2))) * n^(n + 1/2) * exp(n/LambertW(exp(1)*n/2) - n) / LambertW(exp(1)*n/2)^(n + 1/2). - Vaclav Kotesovec, Feb 06 2022

A359659 a(n) = Sum_{k=0..n} k^(k * (n-k+1)).

Original entry on oeis.org

1, 2, 6, 45, 1051, 88602, 27121964, 37004504305, 198705527223757, 5595513387083114570, 686714367475480207331582, 468422339816915120237104999421, 1664212116512828935888786624225704855

Offset: 0

Seiichi Manyama, Jan 10 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..51

Cf. A026898, A349893, A359658.
Cf. A031971, A349836, A349883.
Cf. A003101, A349882.

PARI
```
a(n) = sum(k=0, n, k^(k*(n-k+1)));
```

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

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

G.f.: Sum_{k>=0} (k * x)^k/(1 - k^k * x).
G.f.: Sum_{k>=0} x^k/(1 - (k+1)^(k+1) * x).
a(n) = A349893(n+1) - 1.

A113170 Ascending descending base exponent transform of odd numbers A005408.

Original entry on oeis.org

1, 4, 33, 376, 5665, 115356, 3014209, 95722288, 3619661121, 161338248820, 8349617508961, 493959321484584, 33041900704133473, 2479933070973253516, 207343189445230918785, 19175058576632809926496, 1949302342535131018462849, 216707770770991401785821668

Offset: 1

Jonathan Vos Post, Jan 06 2006

A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154. The parity of this sequence cycles odd, even, odd, even, ... There is no nontrivial integer fixed point of the transform.

			a(2) = 4 because 1^3 + 3^1 = 1 + 3 = 4.
a(3) = 33 because 1^5 + 3^3 + 5^1 = 1 + 27 + 5 = 33.
a(4) = 406 because 1^7 + 3^5 + 5^3 + 7^1 = 1 + 243 + 125 + 7 = 376.
a(5) = 5665 because 1^9 + 3^7 + 5^5 + 7^3 + 9^1 = 5665.
a(6) = 115356 = 1^11 + 3^9 + 5^7 + 7^5 + 9^3 + 11^1.
a(7) = 3014209 = 1^13 + 3^11 + 5^9 + 7^7 + 9^5 + 11^3 + 13^1.
a(8) = 95722288 = 1^15 + 3^13 + 5^11 + 7^9 + 9^7 + 11^5 + 13^3 + 15^1.
a(9) = 3619661121 = 1^17 + 3^15 + 5^13 + 7^11 + 9^9 + 11^7 + 13^5 + 15^3 + 17^1.
a(10) = 161338248820 = 1^19 + 3^17 + 5^15 + 7^13 + 9^11 + 11^9 + 13^7 + 15^5 + 17^3 + 19^1.

G. C. Greubel, Table of n, a(n) for n = 1..295

Cf. A005408, A113122, A113153, A113154.

Table[Sum[(2 k + 1)^(2 n - 2 k + 1), {k, 1, n}], {n, 0, 10}] + 1 (* G. C. Greubel, May 18 2017 *)

for(n=0,25, print1(1 + sum(k=1,n, (2*k+1)^(2*n-2*k+1)), ", ")) \\ G. C. Greubel, May 18 2017

a(1) = 1. For n>1: a(n) = Sum_{i=1..n} (2n+1)^(2n-i).

A351282 a(n) = Sum_{k=0..n} 3^k * k^(n-k).

Original entry on oeis.org

1, 3, 12, 48, 201, 885, 4116, 20298, 106365, 592455, 3503532, 21946620, 145210305, 1011726417, 7400390052, 56668826118, 453116188821, 3774297532467, 32682069679548, 293632972911048, 2732593851548985, 26299137526992525, 261387306941467188, 2679392140776188706

Offset: 0

Vaclav Kotesovec, Feb 06 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..572

Cf. A003101, A351279.

Join[{1}, Table[Sum[3^k*k^(n-k), {k, 0, n}], {n, 1, 25}]]

a(n) = sum(k=0, n, 3^k*k^(n-k)); \\ Michel Marcus, Feb 06 2022

a(n) ~ sqrt(2*Pi/(1 + LambertW(exp(1)*n/3))) * n^(n + 1/2) * exp(n/LambertW(exp(1)*n/3) - n) / LambertW(exp(1)*n/3)^(n + 1/2).

G.f.: Sum_{k>=0} 3^k * x^k / (1 - k*x). - Ilya Gutkovskiy, Feb 06 2022

A121945 a(n) is the sum of the first n factorials in decreasing powers from n to 1. a(n) = Sum_{k = 1..n} k!^(n-k+1).

Original entry on oeis.org

1, 3, 11, 69, 929, 30273, 2591057, 614059329, 423463272449, 907403624202753, 6082394749206781697, 140440480114401911810049, 10845109029138237198786147329, 3088811811740393517911301490890753, 3220352134317904958924570965080200574977, 12657255883388612328426763834234183884771442689

Offset: 1

Tanya Khovanova, Sep 03 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..58

Similar to A003101 = Sum_{k = 1..n} (n-k+1)^k - only with inserted factorials.

List([1..20], n-> Sum([1..n], j-> Factorial(j)^(n-j+1)) ); # G. C. Greubel, Oct 07 2019

[(&+[Factorial(j)^(n-j+1): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Oct 07 2019

seq(add(factorial(j)^(n-j+1), j=1..n), n=1..20); # G. C. Greubel, Oct 07 2019

Table[Sum[Factorial[i]^(n-i+1), {i, n}], {n, 20}]

vector(20, n, sum(j=1, n, (j!)^(n-j+1)) ) \\ G. C. Greubel, Oct 07 2019

[sum(factorial(j)^(n-j+1) for j in (1..n)) for n in (1..20)] # G. C. Greubel, Oct 07 2019

More terms from G. C. Greubel, Oct 07 2019

A130425 a(n) = numerator of Sum_{k=1..n} 1/k^(n+1-k).

Original entry on oeis.org

1, 3, 19, 107, 1471, 164059, 65442581, 26560388929, 10901416818161, 4504891039128649, 20562691778919031051, 94108143760454361244249, 5609165278757040127506253363, 334755533004517896353486403105731

Offset: 1

Leroy Quet, May 26 2007

Cf. A130426, A003101.

Table[Numerator[Sum[1/k^(n + 1 - k), {k, 1, n}]], {n, 1, 20}] (* Stefan Steinerberger, May 30 2007 *)

More terms from Stefan Steinerberger, May 30 2007

A130426 a(n) = denominator of Sum_{k=1..n} 1/k^(n+1-k).

Original entry on oeis.org

1, 2, 12, 72, 1080, 129600, 54432000, 22861440000, 9601804800000, 4032758016000000, 18631342033920000000, 86076800196710400000000, 5169772619814426624000000000, 310496543546054463037440000000000

Offset: 1

Leroy Quet, May 26 2007

Cf. A130425, A003101.

Table[Denominator[Sum[1/k^(n + 1 - k), {k, 1, n}]], {n, 1, 20}] (* Stefan Steinerberger, May 30 2007 *)

More terms from Stefan Steinerberger, May 30 2007

A341436 Numbers k such that k divides Sum_{j=1..k} j^(k+1-j).

Original entry on oeis.org

1, 5, 16, 208, 688, 784, 2864, 9555, 17776, 81239

Offset: 1

Seiichi Manyama, Feb 11 2021

Numbers k such that k divides A003101(k).

a(11) > 10^5.

			1^5 + 2^4 + 3^3 + 4^2 + 5^1 = 65 = 5 * 13. So 5 is a term.

Mathematics Stack Exchange, Is it true that if p != 5 is a prime number then 1^p + 2^(p-1) + ... + (p-1)^2 + p^1 != 0 (mod p)?.

Cf. A003101, A128981, A188775, A341437.

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

isok(n) = sum(k=1, n, Mod(k, n)^(n+1-k))==0;

cp's OEIS Frontend

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

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A349882 Expansion of Sum_{k>=0} k^2 * x^k/(1 - k^2 * x).

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A351279 a(n) = Sum_{k=0..n} 2^k * k^(n-k).

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A359659 a(n) = Sum_{k=0..n} k^(k * (n-k+1)).

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A113170 Ascending descending base exponent transform of odd numbers A005408.

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A351282 a(n) = Sum_{k=0..n} 3^k * k^(n-k).

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A121945 a(n) is the sum of the first n factorials in decreasing powers from n to 1. a(n) = Sum_{k = 1..n} k!^(n-k+1).

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