A215083 -id:A215083 - cp's OEIS frontend

A023002 Sum of 10th powers.

0, 1, 1025, 60074, 1108650, 10874275, 71340451, 353815700, 1427557524, 4914341925, 14914341925, 40851766526, 102769130750, 240627622599, 529882277575, 1106532668200, 2206044295976, 4222038196425, 7792505423049, 13923571680850

Offset: 0

David W. Wilson

T. D. Noe, Table of n, a(n) for n = 0..1000
Bruno Berselli, A description of the recursive method in Formula lines (second formula): website Matem@ticamente (in Italian).
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
Eric Weisstein's World of Mathematics, Power Sum.
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-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), this sequence (m=10), A123095 (m=11), A123094 (m=12), A181134 (m=13).
Row 10 of array A103438.
Cf. A215083, A069095.

[&+[n^10: n in [0..m]]: m in [0..19]];  // Bruno Berselli, Aug 23 2011

A023002:= n-> bernoulli(11, n+1)/11; seq(A023002(n), n=0..30); # G. C. Greubel, Jul 21 2021

Table[Sum[k^10, {k, n}], {n, 0, 30}] (* Vladimir Joseph Stephan Orlovsky, Aug 14 2008 *)
Accumulate[Range[0,20]^10] (* Harvey P. Dale, Aug 23 2011 *)

a(n)=(6*x^11+33*x^10+55*x^9-66*x^7+66*x^5-33*x^3+5*x)/66 \\ Charles R Greathouse IV, Aug 23 2011

a(n)=sum(i=0,10,binomial(11,i)*bernfrac(i)*n^(11-i))/11+n^10 \\ Charles R Greathouse IV, Aug 23 2011

A023002_list, m = [0], [3628800, -16329600, 30240000, -29635200, 16435440, -5103000, 818520, -55980, 1022, -1, 0 , 0]
for _ in range(20):
    for i in range(11):
        m[i+1]+= m[i]
    A023002_list.append(m[-1])
print(A023002_list) # Chai Wah Wu, Nov 05 2014

[bernoulli_polynomial(n,11)/11 for n in range(2, 21)]# Zerinvary Lajos, May 17 2009

a(n) = n*(n+1)*(2*n+1)*(n^2+n-1)(3*n^6 +9*n^5 +2*n^4 -11*n^3 +3*n^2 +10*n -5)/66 (see MathWorld, Power Sum, formula 40). - Bruno Berselli, Apr 26 2010
a(n) = n*A007487(n) - Sum_{i=0..n-1} A007487(i). - Bruno Berselli, Apr 27 2010
From Bruno Berselli, Aug 23 2011: (Start)
a(n) = -a(-n-1).
G.f.: x*(1+x)*(1 +1012*x +46828*x^2 +408364*x^3 +901990*x^4 +408364*x^5 +46828*x^6 +1012*x^7 +x^8)/(1-x)^12. (End)
a(n) = (-1)*Sum_{j=1..10} j*Stirling1(n+1,n+1-j)*Stirling2(n+10-j,n). - Mircea Merca, Jan 25 2014
a(n) = Sum_{i=1..n} J_10(i)*floor(n/i), where J_10 is A069095. - Ridouane Oudra, Jul 17 2025

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

A215077 Binomial convolution of sum of consecutive powers.

Original entry on oeis.org

0, 1, 7, 66, 852, 14020, 280472, 6609232, 179317056, 5505532992, 188717617280, 7143999854464, 296013377405440, 13325516967972352, 647610246703508480, 33794224057227356160, 1884620857353101983744, 111857608180484932648960, 7040178644779119413723136, 468349192560992552808841216, 32836927387372039917034405888

Offset: 0

Olivier Gérard, Aug 02 2012

a(0) could alternatively be defined as 1 from the formula or the convention for 0^0.

This sum is remarkable for its three different decompositions involving powers and binomials (see formulas and cross-refs).

Winston de Greef, Table of n, a(n) for n = 0..372 (first 201 terms from Vincenzo Librandi)

Row sums of A215078, A215079, A215080.

A215084 a(n) = sum of the sums of the k first n-th powers.

Original entry on oeis.org

0, 1, 6, 46, 470, 6035, 93436, 1695036, 35277012, 828707925, 21693441550, 626254969978, 19766667410282, 677231901484775, 25031756512858200, 992872579254244088, 42066929594261568840, 1896157095455962952169, 90601933352843530354170, 4574495282686422755339734, 243359175218492577008763726

Offset: 0

Olivier Gérard, Aug 02 2012

nonn

First term a(0) may be computed as 1 by starting the inner sum at j=0 and taking the convention 0^0 = 1.

			a(3) = (1^3) + (1^3 + 2^3) + (1^3 + 2^3 + 3^3) = (1^3 + 1^3 + 1^3) + (2^3 + 2^3) + (3^3) = 3 * 1^3 + 2 * 2^3 + 1 * 3^3 = 46. - _David A. Corneth_, Jun 27 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Row sums of A215083.

Cf. A086787, A349116.

Table[Sum[Sum[j^n, {j, 1, k}], {k, 0, n}], {n, 0, 20}]
a[n_] := (n+1)*HarmonicNumber[-1, -n] - HarmonicNumber[n, -n-1] + (n+1)*HarmonicNumber[n, -n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 05 2013 *)
Table[Total[Accumulate[Range[n]^n]],{n,0,20}] (* Harvey P. Dale, Mar 29 2020 *)

a(n) = sum(k=1, n, sum(j=1, k, j^n)); \\ Michel Marcus, Jun 25 2018

a(n) = sum(i=1, n, (n+1-i) * i^n); \\ David A. Corneth, Jun 27 2018

a(n) = Sum_{k=1..n} Sum_{j=1..k} j^n.
a(n) = Sum_{k=1..n} H_k^{-n} where H_k^{-n} is the k-th harmonic number of order -n.
a(n) = Sum_{k=1..n} (B(n+1, k+1) - B(n+1, 1))/(n+1), where B(n, x) are the Bernoulli polynomials. - Daniel Suteu, Jun 25 2018
G.f.: Sum_{k>=1} k^k*x^k/(1 - k*x)^2. - Ilya Gutkovskiy, Oct 11 2018
a(n) ~ c * n^n, where c = 1/(1 - 2*exp(-1) + exp(-2)) = 2.50265030107711874333... - Vaclav Kotesovec, Nov 06 2021

A215078 Triangle of sums of the first k n-th powers multiplied by binomial(n,k), read by rows.

Original entry on oeis.org

0, 0, 1, 0, 2, 5, 0, 3, 27, 36, 0, 4, 102, 392, 354, 0, 5, 330, 2760, 6500, 4425, 0, 6, 975, 15880, 73350, 123090, 67171, 0, 7, 2709, 81060, 654500, 2033325, 2637327, 1200304, 0, 8, 7196, 381808, 5064780, 25926824, 59992660, 63259168, 24684612, 0, 9, 18468, 1696464, 35574840, 281668590, 1034305524, 1896003648, 1681960464, 574304985, 0, 10, 46125, 7208880, 232816500, 2740317300, 14981494710, 42457884000, 64240088580, 49143419250, 14914341925

Offset: 0

Olivier Gérard, Aug 02 2012

If one starts the sum at j=0, the initial term T(0,0) is 1.

			  0
  0  1
  0  2    5
  0  3   27       36
  0  4  102      392      354
  0  5  330     2760     6500     4425
  0  6  975    15880    73350   123090    67171
  0  7 2709    81060   654500  2033325  2637327  1200304

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Binomial convolution of A215083.

Cf. A215077 (row sums), A031971 (diagonal)

A215078 := proc(n,k)
        binomial(n,k)*add(j^n,j=1..k) ;
end proc:
seq(seq(A215078(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Jan 27 2023

Flatten[Table[Table[Sum[j^n, {j, 1, k}]*Binomial[n, k], {k, 0, n}], {n, 0, 10}], 1]

T(n,k) = binomial(n,k)*sum(j^n, j=1..k)

A292900 Triangle read by rows, a generalization of the Bernoulli numbers, the numerators for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 1, -4, 0, 0, -1, 47, -10, -1, 0, 1, -221, 205, -209, 0, 0, -1, 953, -5495, 10789, -427, 1, 0, 1, -3953, 123445, -8646163, 177093, -22807, 0, 0, -1, 16097, -2534735, 22337747, -356249173, 3440131, -46212, -1

Offset: 0

Peter Luschny, Oct 01 2017

The diagonal B(n, n) gives the Bernoulli numbers B_n = B_n(1). The formula is due to L. Kronecker and the generalization to Fukuhara, Kawazumi and Kuno.

			The triangle T(n, k) begins:
[0], 1
[1], 0,  1
[2], 0, -1,     1
[3], 0,  1,    -4,        0
[4], 0, -1,    47,      -10,       -1
[5], 0,  1,  -221,      205,     -209,          0
[6], 0, -1,   953,    -5495,    10789,       -427,       1
[7], 0,  1, -3953,   123445, -8646163,     177093,  -22807,      0
[8], 0, -1, 16097, -2534735, 22337747, -356249173, 3440131, -46212, -1
The rational triangle B(n, k) begins:
[0], 1
[1], 0,  1/2
[2], 0, -1/2,      1/6
[3], 0,  1/2,     -4/3,          0
[4], 0, -1/2,    47/12,      -10/3,         -1/30
[5], 0,  1/2,  -221/24,      205/9,       -209/20,          0
[6], 0, -1/2,   953/48,   -5495/54,      10789/80,    -427/10,       1/42
[7], 0,  1/2, -3953/96, 123445/324, -8646163/8640, 177093/200, -22807/105, 0

S. Fukuhara, N. Kawazumi and Y. Kuno, Generalized Kronecker formula for Bernoulli numbers and self-intersections of curves on a surface, arXiv:1505.04840 [math.NT], 2015.
L. Kronecker, Über die Bernoullischen Zahlen, J. Reine Angew. Math. 94 (1883), 268-269.

Cf. A292901 (denominators), B(n, n) = A164555(n)/A027642(n), A215083.

B := (n, k) -> `if`(n = 0, 1, add(((-1)^(j-n)/(j+1))*binomial(k+1, j+1)*add(i^n*(j-i+1)^(k-n), i=0..j), j=0..k)):
for n from 0 to 8 do seq(numer(B(n,k)), k=0..n) od;

B[0, 0] = 1; B[n_, k_] := Sum[(-1)^(j-n)/(j+1)*Binomial[k+1, j+1]* Sum[i^n*(j-i+1)^(k-n) , {i, 0, j}] , {j, 0, k}];
Table[B[n, k] // Numerator, {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 19 2018, from Maple *)

B(n, k) = Sum_{j=0..k}(((-1)^(j-n)/(j+1))*binomial(k+1, j+1)*Sum_{i=0..j}(i^n*(j-i+1)^(k-n))) if n >= 1 and B(0, 0) = 1.
B_n = B(n, n) = Sum_{j=0..n}((-1)^(n-j)/(j+1))*binomial(n+1,j+1)*(Sum_{i=0..j}i^n).
T(n, k) = numerator(B(n, k)).

A367416 Triangle read by rows: T(n,k) = number of solutions to +- 1^k +- 2^k +- 3^k +- ... +- n^k is a k-th power, n >= 2.

Original entry on oeis.org

4, 8, 1, 16, 1, 32, 0, 2, 64, 6, 128, 8, 256, 16, 4, 512, 26, 1024, 17, 10, 2048, 67, 4, 3, 4096, 100, 10, 8192, 137, 34, 6, 16384, 426, 28, 1, 32768, 661, 96, 6, 65536, 1351, 146, 16, 8, 131072, 2637, 230, 15, 262144, 3831, 258, 40, 524288, 8095, 1130, 50

Offset: 2

Jean-Marc Rebert, Jan 26 2024

In the case of n = 1, there are solutions for all k. In particular, 1^k is always a k-th power and -(1^k) is a k-th power for odd k. As a formula: T(1,k) = 1 + (k mod 2). This row is not included in the sequence.

			Triangle begins:
            k = 1      2     3   4  5
  n= 2:         4;
  n= 3:         8,     1;
  n= 4:        16,     1;
  n= 5:        32,     0,    2;
  n= 6:        64,     6;
  n= 7:       128,     8;
  n= 8:       256,    16,    4;
  n= 9:       512,    26;
  n=10:      1024,    17,   10;
  n=11:      2048,    67,    4,  3;
  n=12:      4096,   100,   10;
  n=13:      8192,   137,   34,  6;
  n=14:     16384,   426,   28,  1;
  n=15:     32768,   661,   96,  6;
  n=16:     65536,  1351,  146, 16, 8;
  n=17:    131072,  2637,  230, 15;
  n=18:    262144,  3831,  258, 40;
  n=19:    524288,  8095, 1130, 50;
  n=20:   1048576, 15241,  854, 77, 6;
  ...
The T(6,2) = 6 solutions are:
  - 1^2 - 2^2 + 3^2 - 4^2 + 5^2 + 6^2 = 49 = 7^2,
  - 1^2 - 2^2 + 3^2 + 4^2 + 5^2 - 6^2 =  9 = 3^2,
  - 1^2 - 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 81 = 9^2,
  + 1^2 - 2^2 + 3^2 - 4^2 - 5^2 + 6^2 =  1 = 1^2,
  + 1^2 + 2^2 - 3^2 + 4^2 + 5^2 - 6^2 =  1 = 1^2,
  + 1^2 + 2^2 + 3^2 - 4^2 - 5^2 + 6^2 =  9 = 3^2.

Cf. A063890, A215083, A368243, A368845, A369629.

f(k,u)=my(x=0,v=vector(#u));for(i=1,#u,u[i]=if(u[i]==0,-1,1);v[i]=i^k);u*v~
is(k,u)=my(x=f(k,u));ispower(x,k)
T(n,k)=my(u=vector(n,i,[0,1]),nbsol=0);if(k%2==1,u[1]=[1,1]);forvec(X=u,if(is(k,X),nbsol++));if(k%2==1,nbsol*=2);nbsol

cp's OEIS Frontend

A023002 Sum of 10th powers.

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PARI

Python

Sage

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A123094 Sum of first n 12th powers.

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Mathematica

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Sage

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A123095 Sum of first n 11th powers.

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Magma

Maple

Mathematica

Python

Sage

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

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Magma

Maple

Mathematica

Python

Sage

Formula

A215077 Binomial convolution of sum of consecutive powers.

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Mathematica

PARI

PARI

Formula

A215084 a(n) = sum of the sums of the k first n-th powers.

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Mathematica

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PARI