A076015 -id:A076015 - cp's OEIS frontend

A103438 Square array T(m,n) read by antidiagonals: Sum_{k=1..n} k^m.

0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 5, 6, 4, 0, 1, 9, 14, 10, 5, 0, 1, 17, 36, 30, 15, 6, 0, 1, 33, 98, 100, 55, 21, 7, 0, 1, 65, 276, 354, 225, 91, 28, 8, 0, 1, 129, 794, 1300, 979, 441, 140, 36, 9, 0, 1, 257, 2316, 4890, 4425, 2275, 784, 204, 45, 10

Offset: 0

Ralf Stephan, Feb 11 2005

For the o.g.f.s of the column sequences for this array, see A196837 and the link given there. - Wolfdieter Lang, Oct 15 2011
T(m,n)/n is the m-th moment of the discrete uniform distribution on {1,2,...,n}. - Geoffrey Critzer, Dec 31 2018
T(1,n) divides T(m,n) for odd m. - Franz Vrabec, Dec 23 2020

			Square array begins:
  0, 1,  2,   3,    4,     5,     6,      7,      8,      9, ... A001477;
  0, 1,  3,   6,   10,    15,    21,     28,     36,     45, ... A000217;
  0, 1,  5,  14,   30,    55,    91,    140,    204,    285, ... A000330;
  0, 1,  9,  36,  100,   225,   441,    784,   1296,   2025, ... A000537;
  0, 1, 17,  98,  354,   979,  2275,   4676,   8772,  15333, ... A000538;
  0, 1, 33, 276, 1300,  4425, 12201,  29008,  61776, 120825, ... A000539;
  0, 1, 65, 794, 4890, 20515, 67171, 184820, 446964, 978405, ... A000540;
Antidiagonal triangle begins as:
  0;
  0, 1;
  0, 1,  2;
  0, 1,  3,  3;
  0, 1,  5,  6,  4;
  0, 1,  9, 14, 10,  5;
  0, 1, 17, 36, 30, 15, 6;

J. Faulhaber, Academia Algebrae, Darinnen die miraculosische inventiones zu den höchsten Cossen weiters continuirt und profitirt werden, Augspurg, bey Johann Ulrich Schönigs, 1631.

G. C. Greubel, Antidiagonals n = 0..50, flattened
José L. Cereceda, Sums of powers of integers and hyperharmonic numbers, arXiv:2005.03407 [math.NT], 2020.
T. A. Gulliver, Divisibility of sums of powers of odd integers, Int. Math. For. 5 (2010) 3059-3066.
T. A. Gulliver, Sums of Powers of Integers Divisible by Three, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 38, pp. 1895-1901. - From _N. J. A. Sloane_, Dec 22 2012
V. J. W. Guo and J. Zeng, A q-analogue of Faulhaber's formula for sums of powers, arXiv:math/0501441 [math.CO], 2005.
H. Helfgott and I. M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.
T. Kim, q-analogues of the sums of powers of consecutive integers, arXiv:math/0502113 [math.NT], 2005.
D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comp. 61 (1993), no. 203, 277-294.
Eric Weisstein's World of Mathematics, Discrete Uniform Distribution.
Wikipedia, Faulhaber's formula

Rows include A000027, A000217, A000330, A000537, A000538, A000539, A000540, A000541, A000542, A007487, A023002.
Columns include A000051, A001550, A001551, A001552, A001553, A001554, A001555, A001556, A001557.
Diagonals include A076015 and A031971.
Antidiagonal sums are in A103439.
Antidiagonals are the rows of triangle A192001.

T:= func< n,k | n eq 0 select k else (&+[j^n: j in [0..k]]) >;
[T(n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 22 2021

seq(print(seq(Zeta(0,-k,1)-Zeta(0,-k,n+1),n=0..9)),k=0..6);
# (Produces the square array from the example.) Peter Luschny, Nov 16 2008
# alternative
A103438 := proc(m,n)
    (bernoulli(m+1,n+1)-bernoulli(m+1))/(m+1) ;
    if m = 0 then
        %-1 ;
    else
        % ;
    end if;
end proc: # R. J. Mathar, May 10 2013
# simpler:
A103438 := proc(m,n)
    (bernoulli(m+1,n+1)-bernoulli(m+1,1))/(m+1) ;
end proc: # Peter Luschny, Mar 20 2024

T[m_, n_]:= HarmonicNumber[m, -n]; Flatten[Table[T[m-n, n], {m, 0, 11}, {n, m, 0, -1}]] (* Jean-François Alcover, May 11 2012 *)

PARI
```
T(m,n)=sum(k=0,n,k^m)
```

from itertools import count, islice
from math import comb
from fractions import Fraction
from sympy import bernoulli
def A103438_T(m,n): return sum(k**m for k in range(1,n+1)) if n<=m else int(sum(comb(m+1,i)*(bernoulli(i) if i!=1 else Fraction(1,2))*n**(m-i+1) for i in range(m+1))/(m+1))
def A103438_gen(): # generator of terms
    for m in count(0):
        for n in range(m+1):
            yield A103438_T(m-n,n)
A103438_list = list(islice(A103438_gen(),100)) # Chai Wah Wu, Oct 23 2024

def T(n,k): return (bernoulli_polynomial(k+1, n+1) - bernoulli_polynomial(1, n+1)) /(n+1)
flatten([[T(n-k,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Dec 22 2021

E.g.f.: e^x*(e^(x*y)-1)/(e^x-1).
T(m, n) = Zeta(-n, 1) - Zeta(-n, m + 1), for m >= 0 and n >= 0, where Zeta(z, v) is the Hurwitz zeta function. - Peter Luschny, Nov 16 2008
T(m, n) = HarmonicNumber(m, -n). - Jean-François Alcover, May 11 2012
T(m, n) = (Bernoulli(m + 1, n + 1) - Bernoulli(m + 1, 1)) / (m + 1). - Peter Luschny, Mar 20 2024
T(m, n) = Sum_{k=0...m-n} B(k)*(-1)^k*binomial(m-n,k)*n^(m-n-k+1)/(m-n-k+1), where B(k) = Bernoulli number A027641(k) / A027642(k). - Robert B Fowler, Aug 20 2024
T(m, n) = Sum_{i=1..n} J_m(i)*floor(n/i), where J_m is the m-th Jordan totient function. - Ridouane Oudra, Jul 19 2025

A076014 Triangle in which m-th entry of n-th row is m^(n-1).

Original entry on oeis.org

1, 1, 2, 1, 4, 9, 1, 8, 27, 64, 1, 16, 81, 256, 625, 1, 32, 243, 1024, 3125, 7776, 1, 64, 729, 4096, 15625, 46656, 117649, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721

Offset: 1

Wolfdieter Lang, Oct 02 2002

This becomes triangle A009998(n-1, m-1), n >= m >= 1, if the m-th column entries are divided by m^(m-1).
Row sums give A076015. The m-th column (without leading zeros) gives (m^(m-1)) powers of m, m >= 1.
T(n,m) is the number of functions f:[n-1]->[(n-1)m] such that f(x)=k*x for some positive integer k <= m. Since there exactly m choices for each of the (n-1) images under f, we obtain T(n,m) = m^(n-1). - Dennis P. Walsh, Feb 27 2013
T(n+1,m+1) = (m+1)^n is the number of partial functions from an n-element set to an m-element set, n >= m >= 0. - Mohammad K. Azarian, Jun 28 2021

			For example, T(3,2)=4 since there are exactly 4 functions f from {1,2} to {1,2,3,4} that satisfy f(x)=x or f(x)=2x. If we specify each function by the ordered pair (f(1),f(2)), the four functions are (1,2), (1,4), (2,2), and (2,4). - _Dennis P. Walsh_, Feb 27 2013
Triangle begins:
  1;
  1,   2;
  1,   4,    9;
  1,   8,   27,    64;
  1,  16,   81,   256,   625;
  1,  32,  243,  1024,  3125,   7776;
  1,  64,  729,  4096, 15625,  46656, 117649;
  1, 128, 2187, 16384, 78125, 279936, 823543, 2097152;
  ...

Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. See Theorem 2.5, p. 132.

Cf. A009998, A008279, A008277 (Stirling2).

Cf. A089072.

seq(seq(m^(n-1),m=1..n),n=1..20); # Dennis P. Walsh, Feb 27 2013

Table[m^(n-1),{n,10},{m,n}]//Flatten (* Harvey P. Dale, May 27 2017 *)

T(n, m) = m^(n-1), n >= m >= 1, otherwise 0.
G.f. for m-th column: (m^(m-1))(x^m)/(1-m*x), m >= 1.
a(n,m) = Sum_{p=1..m} Stirling2(n,p)*A008279(m-1, p-1), n >= m >= 1, otherwise 0.

A125120 Sum of values of repunits of length n in base b representation with 1

Original entry on oeis.org

1, 7, 41, 296, 2829, 34637, 519049, 9197344, 188039787, 4356087231, 112754069273, 3224945523736, 100999970565337, 3437517630509497, 126332966608699441, 4986057436997869696, 210331809309776028055, 9443995455881145458715

Offset: 1

Reinhard Zumkeller, Nov 21 2006

			a(4) = [1111]_2 + [1111]_3 + [1111]_4 + [1111]_5 = ((2+1)*2+1)*2+1 + ((3+1)*3+1)*3+1 + ((4+1)*4+1)*4+1 + ((5+1)*5+1)*5+1 = 296.

G. C. Greubel, Table of n, a(n) for n = 1..385
Eric Weisstein's World of Mathematics, Repunit

Row sums of A125118.

Cf. A076015, A228275.

[(&+[(k^n -1)/(k-1): k in [2..n+1]]) : n in [1..30]]; // G. C. Greubel, Aug 14 2022

Table[Sum[(k^n -1)/(k-1), {k, 2, n+1}], {n, 30}] (* G. C. Greubel, Aug 14 2022 *)

a(n) = sum(k=1, n, sum(i=0, n-1, (k+1)^i)); \\ Michel Marcus, Dec 14 2020

[sum(((k+1)^n -1)/k for k in (1..n)) for n in (1..30)] # G. C. Greubel, Aug 14 2022

a(n) = Sum_{k=1..n} Sum_{i=0..n-1} (k+1)^i. [Corrected by Mathew Englander, Dec 14 2020]
a(n) = Sum_{k=1..n} A125118(n,k).
a(n+1) - a(n) = A076015(n+1) + A228275(n+2, n). - Mathew Englander, Dec 14 2020
a(n) = Sum_{j=2..n+1} (j^n - 1)/(j-1)

A284759 a(n) = (Sum_{i=1..n-1} i^(n-2)) mod n^3.

Original entry on oeis.org

0, 1, 3, 14, 100, 115, 196, 500, 189, 333, 847, 1022, 1352, 1671, 1920, 3432, 3757, 2937, 1444, 7730, 1092, 427, 4232, 8668, 15000, 13037, 19197, 20902, 1682, 17999, 16337, 27856, 32043, 31873, 16170, 14298, 47915, 5603, 12792, 8260, 16810, 18949, 51772, 64526

Offset: 1

Felix Fröhlich, Apr 02 2017

nonn

Conjecture: For n > 1, a(n) = 0 if and only if n is a term of A088164, i.e., n is a Wolstenholme prime (cf. Mestrovic, 2012, Conjecture 2.10).

Seiichi Manyama, Table of n, a(n) for n = 1..1000
R. Mestrovic, A congruence modulo n^3 involving two consecutive sums of powers and its applications, arXiv:1211.4570 [math.NT], 2012.

Cf. A000578, A076015, A088164, A284760.

seq(add(i^(n-2),i=1..n-1) mod n^3, n=1..100);

Table[Mod[Sum[i^(n - 2), {i, n - 1}], n^3], {n, 44}] (* Michael De Vlieger, Apr 02 2017 *)

a(n) = lift(Mod(sum(i=1, n-1, i^(n-2)), n^3))

a(n)=my(m=n^3,e=n-2); lift(sum(i=1,n-1, Mod(i,m)^e)) \\ Charles R Greathouse IV, Apr 07 2017

a(n) = A076015(n-1) modulo A000578(n).

A284760 a(n) = Sum_{i=1..n-1}(i^(n-2)) mod n^4.

Original entry on oeis.org

0, 1, 3, 14, 100, 979, 196, 500, 3834, 1333, 2178, 1022, 16731, 12647, 42420, 23912, 23409, 26265, 15162, 79730, 84441, 21723, 28566, 160732, 280625, 329405, 137295, 569702, 74849, 71999, 463202, 715984, 247665, 31873, 1302420, 574170, 807710, 225091, 1377129

Offset: 1

Felix Fröhlich, Apr 02 2017

nonn

Mestrovic conjectures that a(n) > 0 for all n > 1 (Conjecture 2.11).

			For n=5 the sum is 1^3 + 2^3 + 3^3 + 4^3 = 1 + 8 + 27 + 64 = 100; the modulus is 5^4 = 625. So a(5) = 100 mod 625 = 100. - _Peter Munn_, May 01 2017

Seiichi Manyama, Table of n, a(n) for n = 1..1000
R. Mestrovic, A congruence modulo n^3 involving two consecutive sums of powers and its applications, arXiv:1211.4570 [math.NT], 2012.

Cf. A076015, A284759.

Table[Mod[Sum[i^(n - 2), {i, n - 1}], n^4], {n, 39}] (* Michael De Vlieger, Apr 05 2017 *)

a(n) = lift(Mod(sum(i=1, n-1, i^(n-2)), n^4))

a(n)=my(m=n^4,e=n-2); lift(sum(i=1,n-1, Mod(i,m)^e)) \\ Charles R Greathouse IV, Apr 07 2017

a(n) = A076015(n-1) modulo n^4.

A345106 a(n) = Sum_{k=1..n} k^(n - floor(n/k)).

Original entry on oeis.org

1, 3, 14, 96, 971, 12015, 184286, 3283598, 67676125, 1572527901, 40843114146, 1170338862814, 36718016941445, 1251213685475261, 46033362584427670, 1818364700307111794, 76762441669319061911, 3448793841153099408185, 164309637864524321789042

Offset: 1

Seiichi Manyama, Jun 08 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A076015, A344551, A345107.

a[n_] := Sum[k^(n - Floor[n/k]), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Jun 08 2021 *)

PARI
```
a(n) = sum(k=1, n, k^(n-n\k));
```

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

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

A360172 Sequences of length n in [n] not starting with their minimum value.

Original entry on oeis.org

0, 1, 13, 156, 2146, 34455, 638723, 13479760, 319689156, 8425695015, 244459904085, 7745416087332, 266155064108662, 9860698167427471, 391859875043125895

Offset: 1

Olivier Gérard, Jan 28 2023

Other simple classes of endofunctions are counted by this sequence.

			The 13 sequences not starting with their minimum value for n=3 are
  211, 212, 213, 221, 231, 311, 312, 313, 321, 331, 322, 323, 332.

A076015(n) + A360172(n) = A000312(n).

Table[n^n - Plus @@ Table[i^(n - 1), {i, 1, n}], {n, 1, 15}]

A120490 1 + Sum[ k^(n-1), {k,1,n}].

Original entry on oeis.org

2, 4, 15, 101, 980, 12202, 184821, 3297457, 67731334, 1574304986, 40851766527, 1170684360925, 36720042483592, 1251308658130546, 46034015337733481, 1818399978159990977, 76762718946972480010, 3448810852242967123282

Offset: 1

Alexander Adamchuk, Aug 04 2006

nonn

Prime p divides a(p). Prime p divides a(p-2) for p>3. p^2 divides a(p-2) for prime p=7. p^2 divides a(p^2-2) for prime p except p=3. p^3 divides a(p^2-2) for prime p=7. p^3 divides a(p^3-2) for prime p>3. p^4 divides a(p^3-2) for prime p=7. p^4 divides a(p^4-2) for prime p>3. p^5 divides a(p^3-2) for prime p=7. It appears that p^k divides a(p^k-2) for prime p>3 and 7^(k+1) divides a(7^k-2) for integer k>0.

Cf. A076015.

Table[(1+Sum[k^(n-1),{k,1,n}]),{n,1,23}]

a(n) = 1 + Sum[ k^(n-1), {k,1,n}]. a(n) = 1 + A076015[n].

cp's OEIS Frontend

A103438 Square array T(m,n) read by antidiagonals: Sum_{k=1..n} k^m.

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A076014 Triangle in which m-th entry of n-th row is m^(n-1).

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A125120 Sum of values of repunits of length n in base b representation with 1

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A284759 a(n) = (Sum_{i=1..n-1} i^(n-2)) mod n^3.

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A284760 a(n) = Sum_{i=1..n-1}(i^(n-2)) mod n^4.

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A345106 a(n) = Sum_{k=1..n} k^(n - floor(n/k)).

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