A000541 -id:A000541 - cp's OEIS frontend

A254684 Fifth partial sums of seventh powers (A001015).

1, 133, 2842, 29274, 197400, 1001952, 4137966, 14597934, 45454773, 127861825, 330540028, 795609724, 1801339176, 3867558072, 7926516900, 15591322404, 29566276257, 54259095093, 96674782246, 167695627750, 283882296880

Offset: 1

Luciano Ancora, Feb 12 2015

			First differences:   1, 127, 2060, 14324,  63801, ...  (A152726)
----------------------------------------------------------------------
The seventh powers:  1, 128, 2187, 16384,  78125, ...  (A001015)
----------------------------------------------------------------------
First partial sums:  1, 129, 2316, 18700,  96825, ...  (A000541)
Second partial sums: 1, 130, 2446, 21146, 117971, ...  (A250212)
Third partial sums:  1, 131, 2577, 23723, 141694, ...  (A254641)
Fourth partial sums: 1, 132, 2709, 26432, 168126, ...  (A254646)
Fifth partial sums:  1, 133, 2842, 29274, 197400, ...  (this sequence)

Luciano Ancora, Table of n, a(n) for n = 1..1000
Luciano Ancora, Partial sums of m-th powers with Faulhaber polynomials
Luciano Ancora, Pascal’s triangle and recurrence relations for partial sums of m-th powers
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A000541, A001015, A152726, A250212, A254641, A254646, A254681, A254682, A254683.

Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (- 3 + 5 n + n^2) (- 2 + 5 n + n^2) (5 + 5 n + n^2)/95040, {n,21}] (* or *)
CoefficientList[Series[(- 1 - 120 x - 1191 x^2 - 2416 x^3 - 1191 x^4 - 120 x^5 - x^6)/(-1 + x)^13, {x,0,20}], x]

a(n)=n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(-3+5*n+n^2)*(-2+5*n+n^2)*(5+5*n+n^2)/95040 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (- x - 120*x^2 - 1191*x^3 - 2416*x^4 - 1191*x^5 - 120*x^6 - x^7)/(- 1 + x)^13.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(-3 + 5*n + n^2)*(-2 + 5*n + n^2)*(5 + 5*n + n^2)/95040.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + n^7.

A255181 Third differences of seventh powers (A001015).

Original entry on oeis.org

1, 125, 1806, 10206, 35406, 92526, 201726, 388206, 682206, 1119006, 1738926, 2587326, 3714606, 5176206, 7032606, 9349326, 12196926, 15651006, 19792206, 24706206, 30483726, 37220526, 45017406, 53980206, 64219806, 75852126, 88998126, 103783806, 120340206

Offset: 0

Luciano Ancora, Mar 18 2015

			Third differences:   1, 125, 1806, 10206, 35406, ...  (this sequence)
Second differences:  1, 126, 1932, 12138, 47544, ...  (A255177)
First differences:   1, 127, 2059, 14197, 61741, ...  (A022523)
---------------------------------------------------------------------
The seventh powers:  1, 128, 2187, 16384, 78125, ...  (A001015)
---------------------------------------------------------------------

Lucuano Ancora, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000541, A001015, A022523, A255177.

[1,125] cat [42*(3-10*n+15*n^2-10*n^3+5*n^4): n in [2..30]]; // Vincenzo Librandi, Mar 18 2015

Join[{1, 125}, Table[42 (3 - 10 n + 15 n^2 - 10 n^3 + 5 n^4), {n, 2, 30}]]

G.f.: (1 + 120*x + 1191*x^2 + 2416*x^3 + 1191*x^4 + 120*x^5 + x^6)/(1 - x)^5.
a(n) = 42*(3 - 10*n + 15*n^2 - 10*n^3 + 5*n^4) for n>1, a(0)=1, a(1)=125.
a(n) = A255177(n)-A255177(n-1). - R. J. Mathar, Jul 16 2015

Edited by Bruno Berselli, Mar 19 2015

A123094 Sum of first n 12th powers.

Original entry on oeis.org

0, 1, 4097, 535538, 17312754, 261453379, 2438235715, 16279522916, 84998999652, 367428536133, 1367428536133, 4505856912854, 13421957361110, 36720042483591, 93413954858887, 223160292749512, 504635269460168, 1087257506689929, 2244088888116105, 4457403807182266

Offset: 0

Zerinvary Lajos, Sep 27 2006

T. D. Noe, Table of n, a(n) for n = 0..1000
Bruno Berselli, A description of the recursive method in Formula lines (first formula): website Matem@ticamente (in Italian).

Sequences of the form Sum_{j=0..n} j^m : A000217 (m=1), A000330 (m=2), A000537 (m=3), A000538 (m=4), A000539 (m=5), A000540 (m=6), A000541 (m=7), A000542 (m=8), A007487 (m=9), A023002 (m=10), A123095 (m=11), this sequence (m=12), A181134 (m=13).
Cf. A008456, A215083.
Cf. A008275, A008277.

[(&+[j^12: j in [0..n]]): j in [0..30]]; // G. C. Greubel, Jul 21 2021

Maple
```
[seq(add(i^12, i=1..n), n=0..18)];
```

Table[Sum[k^12, {k, n}], {n, 0, 30}] (* Vladimir Joseph Stephan Orlovsky, Aug 14 2008 *)
Accumulate[Range[0,30]^12]  (* Harvey P. Dale, Apr 26 2011 *)

A123094_list, m = [0], [479001600, -2634508800, 6187104000, -8083152000, 6411968640, -3162075840, 953029440, -165528000, 14676024, -519156, 4094, -1, 0 , 0]
for _ in range(10**2):
    for i in range(13):
        m[i+1]+= m[i]
    A123094_list.append(m[-1]) # Chai Wah Wu, Nov 05 2014

[bernoulli_polynomial(n,13)/13 for n in range(1, 30)] # Zerinvary Lajos, May 17 2009

a(n) = n*A123095(n) - Sum_{i=0..n-1} A123095(i). - Bruno Berselli, Apr 27 2010
a(n) = n * (n+1) * (2*n+1) * (105*n^10 +525*n^9 +525*n^8 -1050*n^7 -1190*n^6 +2310*n^5 +1420*n^4 -3285*n^3 -287*n^2 +2073*n -691)/2730. - Bruno Berselli, Oct 03 2010
a(n) = (-1)*Sum_{j=1..12} j*Stirling1(n+1,n+1-j)*Stirling2(n+12-j,n). - Mircea Merca, Jan 25 2014

A123095 Sum of first n 11th powers.

Original entry on oeis.org

0, 1, 2049, 179196, 4373500, 53201625, 415998681, 2393325424, 10983260016, 42364319625, 142364319625, 427675990236, 1170684360924, 2962844754961, 7012409924625, 15662165784000, 33254351828416, 67526248136049, 131794658215281, 248284917113500, 453084917113500

Offset: 0

Zerinvary Lajos, Sep 27 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Sequences of the form Sum_{j=0..n} j^m : A000217 (m=1), A000330 (m=2), A000537 (m=3), A000538 (m=4), A000539 (m=5), A000540 (m=6), A000541 (m=7), A000542 (m=8), A007487 (m=9), A023002 (m=10), this sequence (m=11), A123094 (m=12), A181134 (m=13).
Cf. A008455, A215083.
Cf. A008275, A008277.

[(&+[j^11: j in [0..n]]): n in [0..30]]; // G. C. Greubel, Jul 21 2021

[seq(add(i^11, i=1..n), n=0..20)];
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^11 od: seq(a[n], n=0..13); # Zerinvary Lajos, Feb 22 2008

Table[Sum[k^11, {k, n}], {n, 0, 30}] (* Vladimir Joseph Stephan Orlovsky, Aug 14 2008 *)
Accumulate[Range[0,20]^11] (* Harvey P. Dale, Sep 17 2021 *)

A123095_list, m = [0], [39916800, -199584000, 419126400, -479001600, 322494480, -129230640, 29607600, -3498000, 171006, -2046, 1, 0 , 0]
for _ in range(10**2):
    for i in range(12):
        m[i+1]+= m[i]
    A123095_list.append(m[-1]) # Chai Wah Wu, Nov 05 2014

[(bernoulli_polynomial(n+1, 12) - bernoulli(12))/12  for n in (0..30)] # G. C. Greubel, Jul 21 2021

a(n) = n*A023002(n) - Sum_{i=0..n-1} A023002(i). - Bruno Berselli, Apr 27 2010
a(n) = n^2*(n+1)^2*(2*n^8 +8*n^7 +4*n^6 -16*n^5 -5*n^4 +26*n^3 -3*n^2 -20*n +10)/24. - Bruno Berselli, Oct 03 2010
G.f.: x*(x^10 +2036*x^9 +152637*x^8 +2203488*x^7 +9738114*x^6 +15724248*x^5 +9738114*x^4 +2203488*x^3 +152637*x^2 +2036*x +1)/(1-x)^13. - Colin Barker, May 27 2012
a(n) = (-1)*Sum_{j=1..11} j*Stirling1(n+1,n+1-j)*Stirling2(n+11-j,n). - Mircea Merca, Jan 25 2014
a(n) = 1728*A006542(n+2)^2 + 216*A288876(n-2) + 96*A006542(n+2) + A000537(n). - Yasser Arath Chavez Reyes, May 25 2024

A181134 Sum of 13th powers: a(n) = Sum_{j=0..n} j^13.

Original entry on oeis.org

0, 1, 8193, 1602516, 68711380, 1289414505, 14350108521, 111239118928, 660994932816, 3202860761145, 13202860761145, 47725572905076, 154718778284148, 457593884876401, 1251308658130545, 3197503726489920

Offset: 0

Bruno Berselli, Oct 05 2010 - Oct 18 2010

This form of recurrence is a general property of the array in A103438 (sums of the first n-th powers).

Bruno Berselli, Table of n, a(n) for n = 0..10000.
Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian), 2008.
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Cf. A010801.
Sequences of the form Sum_{j=0..n} j^m : A000217 (m=1), A000330 (m=2), A000537 (m=3), A000538 (m=4), A000539 (m=5), A000540 (m=6), A000541 (m=7), A000542 (m=8), A007487 (m=9), A023002 (m=10), A123095 (m=11), A123094 (m=12), A181134 (m=13).
Cf. A215083, A253712.

[(&+[j^13: j in [0..n]]): n in [0..30]]; // G. C. Greubel, Jul 21 2021

A181134 := proc(n) (bernoulli(14,n+1) - bernoulli(14))/14 ; end proc: seq(A181134(n), n=0..10); # R. J. Mathar, Oct 14 2010

Accumulate[Range[0,20]^13] (* Harvey P. Dale, Oct 30 2017 *)

A181134_list, m = [0], [6227020800, -37362124800, 97037740800, -142702560000, 130456085760, -76592355840, 28805736960, -6711344640, 901020120, -60780720, 1569750, -8190, 1, 0 , 0]
for _ in range(10**2):
    for i in range(14):
        m[i+1]+= m[i]
    A181134_list.append(m[-1]) # Chai Wah Wu, Nov 06 2014

[(bernoulli_polynomial(n+1, 14) - bernoulli(14))/14  for n in (0..30)] # G. C. Greubel, Jul 21 2021

For n>0, a(n) = n*A123094(n) - Sum_{i=0..n-1} A123094(i), where Sum_{i=0..n-1} A123094(i) = A253712(n-1) = (n-1)*n^2*(n+1)*(30*n^10 - 425*n^8 + 2578*n^6 - 8147*n^4 + 12874*n^2 - 7601)/5460.
a(n) = a(-n-1) = (n*(n + 1))^2*(30*n^10 + 150*n^9 + 125*n^8 - 400*n^7 - 326*n^6 + 1052*n^5 + 367*n^4 - 1786*n^3 + 202*n^2 + 1382*n - 691)/420.
G.f.: see comment of Vladeta Jovovic in A000538.
a(n) = -Sum_{j=1..13} j*Stirling1(n+1,n+1-j)*Stirling2(n+13-j,n). - Mircea Merca, Jan 25 2014

A081175 Numbers of the form Sum_{i=1..k} i^j, j >= 1, k >= 1.

Original entry on oeis.org

1, 3, 5, 6, 9, 10, 14, 15, 17, 21, 28, 30, 33, 36, 45, 55, 65, 66, 78, 91, 98, 100, 105, 120, 129, 136, 140, 153, 171, 190, 204, 210, 225, 231, 253, 257, 276, 285, 300, 325, 351, 354, 378, 385, 406, 435, 441, 465, 496, 506, 513, 528, 561, 595, 630, 650, 666, 703

Offset: 1

N. J. A. Sloane, Apr 18 2003

Union of sums of k-th powers, for k >= 1.

			30 is in the set because 30 = 1^2 + 2^2 + 3^2 + 4^2 (j=2, k=4).

Robert Israel, Table of n, a(n) for n = 1..10000
Michael Penn, an excruciatingly deep dive into the power sum., YouTube video, 2022.

Cf. A000217, A000330, A000537, A000538, A000539, A000540, A000541, A000542, A007487, A023002.

For primes in this sequence see A164307.

N:= 1000: # to get all terms <= N
A:=select(`<=`,{1, seq(seq(sum(i^k,i=1..m), m=2..floor((N*(k+1))^(1/(k+1)))),k = 1 ..ilog2(N-1))},N):
sort(convert(A,list)); # Robert Israel, Jan 26 2015

Take[ Union[ Flatten[ Table[ Sum[ i^j, {i, 1, n}], {j, 1, 9}, {n, 1, 40}]]], 60]

Corrected and extended by Robert G. Wilson v, May 08 2003

A134163 1 + 12n + 81n^3 + n(105n + 81*n^3)/2.

Original entry on oeis.org

1, 187, 1531, 5977, 16441, 36811, 71947, 127681, 210817, 329131, 491371, 707257, 987481, 1343707, 1788571, 2335681, 2999617, 3795931, 4741147, 5852761, 7149241, 8650027, 10375531, 12347137, 14587201, 17119051, 19966987, 23156281, 26713177

Offset: 0

Artur Jasinski, Oct 10 2007

A000541(n) is divisible by A000537(n) if and only n is congruent to 1 mod 3 (see A016777).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000537, A000541, A119617, A134153, A134154, A133180, A134158, A134159, A134160.

[1 + 12*n + 81*n^3 + n*(105*n+ 81*n^3)/2: n in [0..30]]; // Vincenzo Librandi, May 09 2011

A134163:=n->1 + 12*n + 81*n^3 + n*(105*n + 81*n^3)/2: seq(A134163(n), n=0..30); # Wesley Ivan Hurt, Oct 23 2014

Table[(3(3n + 1)^4 + 6(3n + 1)^3 - (3n + 1)^2 - 4 (3n + 1) + 2)/6, {n, 0, 100}] (* or *) Table[Sum[k^7, {k, 1, 3n + 1}]/Sum[k^3, {k, 1, 3n + 1}], {n, 0, 100}]

a(n) = (3(3n + 1)^4 + 6(3n + 1)^3 - (3n + 1)^2 - 4 (3n + 1) + 2)/6.
a(n) = ( sum_{k=1..3n+1} k^7 ) / ( sum_{k=1..3n+1} k^3 ).
G.f.: (1+182*x+606*x^2+182*x^3+x^4)/(1-x)^5. - R. J. Mathar, Nov 14 2007
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Wesley Ivan Hurt, Oct 23 2014

A202107 a(n) = n^4*(n+1)^4/8.

Original entry on oeis.org

2, 162, 2592, 20000, 101250, 388962, 1229312, 3359232, 8201250, 18301250, 37949472, 74030112, 137149922, 243101250, 414720000, 684204032, 1095962562, 1710072162, 2606420000, 3889620000, 5694792642, 8194304162, 11605565952, 16200000000, 22313281250, 30356972802

Offset: 1

Martin Renner, Dec 11 2011

A relation between fourth powers and the sum of fifth and seventh powers. See the first formula, which is from Beiler.

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 161.

Temple Rice Hollcroft, On sums of powers of n consecutive integers, Bulletin of the American Mathematical Society 59 (1953), nr. 6, p. 526 (574t).
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000217, A000539, A000541, A002117, A059977.

A202107:=n->(n^4)*(n+1)^4/8; seq(A202107(n), n=1..100); # Wesley Ivan Hurt, Nov 12 2013

Table[n^4 (n+1)^4/8, {n, 100}] (* Wesley Ivan Hurt, Nov 12 2013 *)

a(n) = 2*(Sum_{k=1..n} k)^4 = Sum_{k=1..n} (k^5 + k^7).
a(n) = 2*A059977(n-1).
a(n) = A000539(n) + A000541(n).
G.f.: -2*x*(1+72*x+603*x^2+1168*x^3+603*x^4+72*x^5+x^6) / (x-1)^9. - R. J. Mathar, Dec 13 2011
a(n) = 2*(A000217(n)^4). - Zak Seidov, Jan 21 2012
From Amiram Eldar, Apr 09 2024: (Start)
Sum_{n>=1} 1/a(n) = 8*Pi^4/45 + 80*Pi^2/3 - 280.
Sum_{n>=1} (-1)^(n+1)/a(n) = 280 - 320*log(2) - 48*zeta(3). (End)

cp's OEIS Frontend

A254684 Fifth partial sums of seventh powers (A001015).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A255181 Third differences of seventh powers (A001015).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A123094 Sum of first n 12th powers.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Python

Sage

Formula

A123095 Sum of first n 11th powers.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Python

Sage

Formula

A181134 Sum of 13th powers: a(n) = Sum_{j=0..n} j^13.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Python

Sage

Formula

A081175 Numbers of the form Sum_{i=1..k} i^j, j >= 1, k >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A134163 1 + 12*n + 81*n^3 + n*(105*n + 81*n^3)/2.

Views

Author

Keywords

A134163 1 + 12n + 81n^3 + n(105n + 81*n^3)/2.