A000542 -id:A000542 - cp's OEIS frontend

A123094 Sum of first n 12th powers.

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

A255182 Third differences of eighth powers (A001016).

Original entry on oeis.org

1, 253, 5796, 46620, 213444, 697788, 1832292, 4131036, 8329860, 15426684, 26721828, 43858332, 68862276, 104183100, 152733924, 217931868, 303738372, 414699516, 555986340, 733435164, 953587908, 1223732412, 1551942756, 1947119580, 2419030404, 2978349948

Offset: 0

Luciano Ancora, Mar 18 2015

			Third differences:   1, 253, 5796, 46620, 213444, ...  (this sequence)
Second differences:  1, 254, 6050, 52670, 266114, ...  (A255178)
First differences:   1, 255, 6305, 58975, 325089, ...  (A022524)
---------------------------------------------------------------------
The seventh powers:  1, 253, 5796, 46620, 213444, ...  (A001016)
---------------------------------------------------------------------

Luciano Ancora, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000542, A001016, A022524, A254642, A255178, A255181.

[1,253] cat [84*(2*n-1)*(2*n^4-4*n^3+8*n^2-6*n+3): n in [2..30]]; // Vincenzo Librandi, Mar 18 2015

Join[{1, 253}, Table[84 (2 n - 1) (2 n^4 - 4 n^3 + 8 n^2 - 6 n + 3), {n, 2, 30}]]

G.f.: (1 + 247*x + 4293*x^2 + 15619*x^3 + 15619*x^4 + 4293*x^5 + 247*x^6 + x^7)/(1 - x)^6.

a(n) = 84*(2*n - 1)*(2*n^4 - 4*n^3 + 8*n^2 - 6*n + 3) for n>1, a(0)=1, a(1)=253.

Edited by Bruno Berselli, Mar 19 2015

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

A261032 a(n) = (-1)^n(n^8 + 4n^7 - 14n^5 + 28n^3 - 17*n)/2.

Original entry on oeis.org

0, -1, 255, -6306, 59230, -331395, 1348221, -4416580, 12360636, -30686085, 69313915, -145044966, 284936730, -530793991, 944995065, -1617895560, 2677071736, -4298685705, 6721274871, -10262288170, 15337711830, -22485147531, 32390726005, -45920259276, 64155054900, -88432835725

Offset: 0

Ilya Gutkovskiy, Nov 18 2015

Alternating sum of eighth powers (A001016).

For n>0, a(n) is divisible by A000217(n).

			a(0) = 0^8 = 0,
a(1) = 0^8 -1^8 = -1,
a(2) = 0^8 -1^8 + 2^8 = 255,
a(3) = 0^8 -1^8 + 2^8 - 3^8 = -6306,
a(4) = 0^8 -1^8 + 2^8 - 3^8 + 4^8 = 59230,
a(5) = 0^8 -1^8 + 2^8 - 3^8 + 4^8 - 5^8 = -331395, etc.

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (-9,-36,-84,-126,-126,-84,-36,-9,-1 ).

Cf. A000217, A001016, A000542, A089594, A232599, A062392, A062393, A152725, A152726.

[(-1)^n*(n^8+4*n^7-14*n^5+28*n^3-17*n)/2: n in [0..30]]; // Vincenzo Librandi, Nov 20 2015

seq((-1)^n*(n^8 + 4*n^7 - 14*n^5 + 28*n^3 - 17*n)/2, n = 0 .. 100); # Robert Israel, Nov 18 2015

Table[(1/2) (-1)^n n (n + 1) (n^6 + 3 n^5 - 3 n^4 - 11 n^3 + 11 n^2 + 17 n - 17), {n, 0, 25}]

vector(100, n, n--; (-1)^n*(n^8+4*n^7-14*n^5+28*n^3-17*n)/2) \\ Altug Alkan, Nov 18 2015

[(-1)^n*(n^8 +4*n^7 -14*n^5 +28*n^3 -17*n)/2 for n in (0..40)] # G. C. Greubel, Apr 02 2021

G.f.: -x*(1 - 246*x + 4047*x^2 - 11572*x^3 + 4047*x^4 - 246*x^5 + x^6)/(1 + x)^9.
a(n) = Sum_{k = 0..n} (-1)^k*k^8.
a(n) = (-1)^n*n*(n + 1)*(n^6 + 3*n^5 - 3*n^4 - 11*n^3 + 11*n^2 + 17*n - 17)/2.
Sum_{n>0} 1/a(n) = -0.9962225712723456482...
Sum_{j=0..9} binomial(9,j)*a(n-j) = 0. - Robert Israel, Nov 18 2015
E.g.f.: (x/2)*(-2 +253*x -1848*x^2 +2961*x^3 -1596*x^4 +350*x^5 -32*x^6 +x^7)*exp(-x). - G. C. Greubel, Apr 02 2021

cp's OEIS Frontend

A123094 Sum of first n 12th powers.

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A123095 Sum of first n 11th powers.

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

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A255182 Third differences of eighth powers (A001016).

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

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A261032 a(n) = (-1)^n*(n^8 + 4*n^7 - 14*n^5 + 28*n^3 - 17*n)/2.

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A261032 a(n) = (-1)^n(n^8 + 4n^7 - 14n^5 + 28n^3 - 17*n)/2.