A254640 -id:A254640 - cp's OEIS frontend

A000538 Sum of fourth powers: 0^4 + 1^4 + ... + n^4.

0, 1, 17, 98, 354, 979, 2275, 4676, 8772, 15333, 25333, 39974, 60710, 89271, 127687, 178312, 243848, 327369, 432345, 562666, 722666, 917147, 1151403, 1431244, 1763020, 2153645, 2610621, 3142062, 3756718, 4463999, 5273999, 6197520, 7246096, 8432017, 9768353

Offset: 0

N. J. A. Sloane

This sequence is related to A000537 by the transform a(n) = n*A000537(n) - Sum_{i=0..n-1} A000537(i). - Bruno Berselli, Apr 26 2010
A formula for the r-th successive summation of k^4, for k = 1 to n, is ((12*n^2+(12*n-5)*r+r^2)*(2*n+r)*(n+r)!)/((r+4)!*(n-1)!), (H. W. Gould). - Gary Detlefs, Jan 02 2014
The number of four dimensional hypercubes in a 4D grid with side lengths n. This applies in general to k dimensions. That is, the number of k-dimensional hypercubes in a k-dimensional grid with side lengths n is equal to the sum of 1^k + 2^k + ... + n^k. - Alejandro Rodriguez, Oct 20 2020

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 222.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1991, p. 275.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
J. L. Bailey, Jr., A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., 2 (1931), 355-359.
J. L. Bailey, A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., 2 (1931), 355-359. [Annotated scanned copy]
Bruno Berselli, A description of the transform in Comments lines: website Matem@ticamente (in Italian).
Stefano Capparelli, Notes on Discrete Math, Società Editrice Esculapio SRL (2019) 3-4.
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.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein, MathWorld: Faulhaber's Formula
Wikipedia, Faulhaber's formula
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000217, A000330, A000537, A000539, A000540, A000541, A000542, A007487, A023002, A064538, A101089.
Row 4 of array A103438.
Cf. A000583.
Cf. A254640.

a000538 n = (3 * n * (n + 1) - 1) * (2 * n + 1) * (n + 1) * n `div` 30
-- Reinhard Zumkeller, Nov 11 2012

[n*(1+n)*(1+2*n)*(-1+3*n+3*n^2)/30: n in [0..35]]; // Vincenzo Librandi, Apr 04 2015

A000538 := n-> n*(n+1)*(2*n+1)*(3*n^2+3*n-1)/30;

Accumulate[Range[0,40]^4] (* Harvey P. Dale, Jan 13 2011 *)
CoefficientList[Series[x (1 + 11 x + 11 x^2 + x^3)/(1 - x)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 07 2015 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 1, 17, 98, 354, 979}, 35] (* Jean-François Alcover, Feb 09 2016 *)
Table[x^5/5+x^4/2+x^3/3-x/30,{x,40}] (* Harvey P. Dale, Jun 06 2021 *)

A000538(n):=n*(n+1)*(2*n+1)*(3*n^2+3*n-1)/30$
makelist(A000538(n),n,0,30); /* Martin Ettl, Nov 12 2012 */

a(n) = n*(1+n)*(1+2*n)*(-1+3*n+3*n^2)/30 \\ Charles R Greathouse IV, Nov 20 2012

concat(0, Vec(x*(1+11*x+11*x^2+x^3)/(1-x)^6 + O(x^100))) \\ Altug Alkan, Dec 07 2015

A000538_list, m = [0], [24, -36, 14, -1, 0, 0]
for _ in range(10**2):
    for i in range(5):
        m[i+1] += m[i]
    A000538_list.append(m[-1]) # Chai Wah Wu, Nov 05 2014

def A000538(n): return n*(n**2*(n*(6*n+15)+10)-1)//30 # Chai Wah Wu, Oct 03 2024

[bernoulli_polynomial(n,5)/5 for n in range(1, 35)] # Zerinvary Lajos, May 17 2009

a(n) = n*(1+n)*(1+2*n)*(-1+3*n+3*n^2)/30.
The preceding formula is due to al-Kachi (1394-1437). - Juri-Stepan Gerasimov, Jul 12 2009
G.f.: x*(x+1)*(1+10*x+x^2)/(1-x)^6. Simon Plouffe in his 1992 dissertation. More generally, the o.g.f. for Sum_{k=0..n} k^m is x*E(m, x)/(1-x)^(m+2), where E(m, x) is the Eulerian polynomial of degree m (cf. A008292). The e.g.f. for these o.g.f.s is: x/(1-x)^2*(exp(y/(1-x))-exp(x*y/(1-x)))/(exp(x*y/(1-x))-x*exp(y/(1-x))). - Vladeta Jovovic, May 08 2002
a(n) = Sum_{i = 1..n} J_4(i)*floor(n/i), where J_4 is A059377. - Enrique Pérez Herrero, Feb 26 2012
a(n) = 5*a(n-1) - 10* a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + 24. - Ant King, Sep 23 2013
a(n) = -Sum_{j=1..4} j*Stirling1(n+1,n+1-j)*Stirling2(n+4-j,n). - Mircea Merca, Jan 25 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = -30*(4 + 3/cos(sqrt(7/3)*Pi/2))*Pi/7. - Vaclav Kotesovec, Feb 13 2015
a(n) = (n + 1)*(n + 1/2)*n*(n + 1/2 + sqrt(7/12))*(n + 1/2  - sqrt(7/12))/5, see the Graham et al. reference, p. 275. - Wolfdieter Lang, Apr 02 2015

The general V. Jovovic formula has been slightly changed after his approval by Wolfdieter Lang, Nov 03 2011

A254647 Fourth partial sums of eighth powers (A001016).

Original entry on oeis.org

1, 260, 7595, 94360, 723534, 4037712, 17944290, 67127880, 219319815, 642251428, 1718012933, 4258676240, 9892043980, 21721707840, 45414150132, 90930820464, 175208925885, 326205634020, 588861675535, 1033717781096, 1769137540730, 2958360418000, 4842936861750, 7774492635000

Offset: 1

Luciano Ancora, Feb 05 2015

			The eighth powers:   1, 256, 6561, 65536, 390625, ... (A001016)
First partial sums:  1, 257, 6818, 72354, 462979, ... (A000542)
Second partial sums: 1, 258, 7076, 79430, 542409, ... (A253636)
Third partial sums:  1, 259, 7335, 86765, 629174, ... (A254642)
Fourth partial sums: 1, 260, 7595, 94360, 723534, ... (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. A000542, A001016, A253636, A254642, A254644, A254645, A254646.

List([1..30], n-> Binomial(n+4,5)*(n+2)*((n+2)^2-3)*(2*(n+2)^4 -28*(n+2)^2 +101)/198); # G. C. Greubel, Aug 28 2019

[Binomial(n+4,5)*(n+2)*((n+2)^2-3)*(2*(n+2)^4 -28*(n+2)^2 +101)/198: n in [1..30]]; // G. C. Greubel, Aug 28 2019

seq(binomial(n+4,5)*(n+2)*((n+2)^2-3)*(2*(n+2)^4 -28*(n+2)^2 +101)/198, n=1..30); # G. C. Greubel, Aug 28 2019

Table[n(1+n)(2+n)^2(3+n)(4+n)(1+4n+n^2)(21 -48n +20n^2 +16n^3 +2n^4 )/23760, {n,20}] (* or *)
Accumulate[Accumulate[Accumulate[Accumulate[Range[20]^8]]]] (* or *)
CoefficientList[Series[(1 +247x +4293x^2 +15619x^3 +15619x^4 +4293x^5 + 247x^6 +x^7)/(1-x)^13, {x,0,19}], x]

a(n)=n*(1+n)*(2+n)^2*(3+n)*(4+n)*(1+4*n+n^2)*(21-48*n+20*n^2 +16*n^3+2*n^4)/23760 \\ Charles R Greathouse IV, Sep 08 2015

vector(30, n, m=n+2; binomial(m+2,5)*m*(m^2-3)*(2*m^4-28*m^2 +101)/198)

[binomial(n+4,5)*(n+2)*((n+2)^2-3)*(2*(n+2)^4 -28*(n+2)^2 +101)/198 for n in (1..30)] # G. C. Greubel, Aug 28 2019

G.f.: x*(1 +247*x +4293*x^2 +15619*x^3 +15619*x^4 +4293*x^5 +247*x^6 +x^7)/(1-x)^13.
a(n) = n*(1+n)*(2+n)^2*(3+n)*(4+n)*(1 +4*n +n^2)*(21 -48*n +20*n^2 + 16*n^3 +2*n^4)/23760.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + n^8.

A101092 Second partial sums of fifth powers (A000584).

Original entry on oeis.org

1, 34, 310, 1610, 6035, 18236, 47244, 109020, 229845, 450670, 832546, 1463254, 2465255, 4005080, 6304280, 9652056, 14419689, 21076890, 30210190, 42543490, 58960891, 80531924, 108539300, 144509300, 190244925, 247861926, 319827834, 409004110, 518691535

Offset: 1

Cecilia Rossiter, Dec 14 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Luciano Ancora, Recurrence relation for the second partial sums of m-th powers
Luciano Ancora, Second partial sums of the m-th powers
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.
C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 1. - _N. J. A. Sloane_, Mar 23 2014 (But beware of a typo)
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Dead link]
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013]
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A000539, A101099.

Cf. A254640.

[(n*(1+n)*(2+n)*(-1+n*(2+n))*(1+2*n*(2+n)))/84: n in [1..40]]; // Vincenzo Librandi, Mar 24 2014

f:=n->(2*n^7-7*n^5+7*n^3-2*n)/84;
[seq(f(n),n=0..50)];  # N. J. A. Sloane, Mar 23 2014

CoefficientList[Series[(1 + 26 x + 66 x^2 + 26 x^3 + x^4)/, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 24 2014 *)
Nest[Accumulate,Range[30]^5,2] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,34,310,1610,6035,18236,47244,109020},30] (* Harvey P. Dale, Aug 22 2022 *)

a(n)=n*(n+1)*(n+2)*(n*(n+2)-1)*(2*n*(2 + n)+1)/84 \\ Charles R Greathouse IV, Apr 21 2015

def A101092(n): return n*(n*(n*(n*(n*(n*(n+7<<1)+35)+35)+7)-7)-2)//84 # Chai Wah Wu, Oct 03 2024

[n*(1+n)*(2+n)*(n*(2+n)-1)*(1+2*n*(2+n))/84 for n in range(1,30)] # Danny Rorabaugh, Apr 21 2015

a(n) = (n*(1 + n)*(2 + n)*(-1 + n*(2 + n))*(1 + 2*n*(2 + n)))/84.
G.f.: x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1-x)^8. - Colin Barker, Apr 16 2012
a(n) = Sum_{i=1..n} i*(n+1-i)^5, by the definition. - Bruno Berselli, Jan 31 2014
a(n) = 2*a(n-1) - a(n-2) + n^5. - Luciano Ancora, Jan 08 2015
E.g.f.: exp(x)*x*(84 + 1344*x + 2954*x^2 + 1995*x^3 + 525*x^4 + 56*x^5 + 2*x^6)/84. - Stefano Spezia, May 04 2024

Edited by Ralf Stephan, Dec 16 2004

A254681 Fifth partial sums of fourth powers (A000583).

Original entry on oeis.org

1, 21, 176, 936, 3750, 12342, 35112, 89232, 207207, 446875, 906048, 1743248, 3206268, 5670588, 9690000, 16062144, 25912029, 40797009, 62837104, 94875000, 140670530, 205134930, 294610680, 417203280, 583171875, 805386231

Offset: 1

Luciano Ancora, Feb 12 2015

			Fourth differences:  1, 12,  23,  24, (repeat 24)  ...   (A101104)
Third differences:   1, 13,  36,  60,   84,   108, ...   (A101103)
Second differences:  1, 14,  50, 110,  194,   302, ...   (A005914)
First differences:   1, 15,  65, 175,  369,   671, ...   (A005917)
-------------------------------------------------------------------------
The fourth powers:   1, 16,  81, 256,  625,  1296, ...   (A000583)
-------------------------------------------------------------------------
First partial sums:  1, 17,  98, 354,  979,  2275, ...   (A000538)
Second partial sums: 1, 18, 116, 470, 1449,  3724, ...   (A101089)
Third partial sums:  1, 19, 135, 605, 2054,  5778, ...   (A101090)
Fourth partial sums: 1, 20, 155, 760, 2814,  8592, ...   (A101091)
Fifth partial sums:  1, 21, 176, 936, 3750, 12342, ...   (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 (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A000538, A000583, A005914, A005917, A101089, A101090, A101091, A101103, A101104, A254682, A254683, A254684.

[Binomial(n+5,6)*n*(n+5)*(2*n+5)/42: n in [1..30]]; // G. C. Greubel, Dec 01 2018

seq(coeff(series((x+11*x^2+11*x^3+x^4)/(1-x)^10,x,n+1), x, n), n = 1 .. 30); # Muniru A Asiru, Dec 02 2018

Table[n^2(1+n)(2+n)(3+n)(4+n)(5+n)^2(5+2n)/30240, {n,26}] (* or *)
CoefficientList[Series[(1 + 11 x + 11 x^2 + x^3)/(1-x)^10, {x,0,25}], x]
CoefficientList[Series[(1/30240)E^x (30240 + 604800 x + 2041200 x^2 + 2368800 x^3 + 1233540 x^4 + 326592 x^5 + 46410 x^6 + 3540 x^7 + 135 x^8 + 2 x^9), {x, 0, 50}], x]*Table[n!, {n, 0, 50}] (* Stefano Spezia, Dec 02 2018 *)
Nest[Accumulate[#]&,Range[30]^4,5] (* Harvey P. Dale, Jan 03 2022 *)

my(x='x+O('x^30)); Vec((x+11*x^2+11*x^3+x^4)/(1-x)^10) \\ G. C. Greubel, Dec 01 2018

[binomial(n+5,6)*n*(n+5)*(2*n+5)/42 for n in (1..30)] # G. C. Greubel, Dec 01 2018

G.f.:(x + 11*x^2 + 11*x^3 + x^4)/(1 - x)^10.
a(n) = n^2*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)^2*(5 + 2*n)/30240.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + n^4.
E.g.f.: (1/30240)*exp(x)*(30240 + 604800*x + 2041200*x^2 + 2368800*x^3 + 1233540*x^4 + 326592*x^5 + 46410*x^6 + 3540*x^7 + 135*x^8 + 2*x^9). - Stefano Spezia, Dec 02 2018
From Amiram Eldar, Jan 26 2022: (Start)
Sum_{n>=1} 1/a(n) = 172032*log(2)/125 - 2382233/2500.
Sum_{n>=1} (-1)^(n+1)/a(n) = 42*Pi^2/25 - 43008*Pi/125 + 2663213/2500. (End)

A254469 Sixth partial sums of cubes (A000578).

Original entry on oeis.org

1, 14, 96, 450, 1650, 5082, 13728, 33462, 75075, 157300, 311168, 586092, 1058148, 1841100, 3100800, 5073684, 8090181, 12603954, 19228000, 28778750, 42329430, 61274070, 87403680, 122996250, 170922375, 234768456, 318979584, 429024376, 571584200, 754769400

Offset: 1

Luciano Ancora, Feb 15 2015

			First differences:   1,  7, 19,  37,   61,   91, ... (A003215)
-------------------------------------------------------------------------
The cubes:           1,  8, 27,  64,  125,  216, ... (A000578)
-------------------------------------------------------------------------
First partial sums:  1,  9, 36, 100,  225,  441, ... (A000537)
Second partial sums: 1, 10, 46, 146,  371,  812, ... (A024166)
Third partial sums:  1, 11, 57, 203,  574, 1386, ... (A101094)
Fourth partial sums: 1, 12, 69, 272,  846, 2232, ... (A101097)
Fifth partial sums:  1, 13, 82, 354, 1200, 3432, ... (A101102)
Sixth partial sums:  1, 14, 96, 450, 1650, 5082, ... (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 .
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. 15.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A000537, A000578, A003215, A024166, A101094, A101097, A101102, A254470, A254471, A254472.

[n*(1+n)^2*(2+n)*(3+n)*(4+n)*(5+n)^2*(6+n)/60480: n in [1..30]]; // Vincenzo Librandi, Feb 15 2015

Table[n (1 + n)^2 (2 + n) (3 + n) (4 + n) (5 + n)^2 (6 + n)/60480, {n, 27}] (* or *) CoefficientList[Series[(1 + 4 x + x^2)/(- 1 + x)^10, {x, 0, 26}], x]
Nest[Accumulate,Range[30]^3,6] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,14,96,450,1650,5082,13728,33462,75075,157300},30] (* Harvey P. Dale, Sep 03 2016 *)

a(n)=n*(1+n)^2*(2+n)*(3+n)*(4+n)*(5+n)^2*(6+n)/60480 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (x + 4*x^2 + x^3)/(- 1 + x)^10.
a(n) = n*(1 + n)^2*(2 + n)*(3 + n)*(4 + n)*(5 + n)^2*(6 + n)/60480.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) + n^3.
From Amiram Eldar, Jan 26 2022: (Start)
Sum_{n>=1} 1/a(n) = 217/200.
Sum_{n>=1} (-1)^(n+1)/a(n) = 223769/200 - 8064*log(2)/5. (End)

A254644 Fourth partial sums of fifth powers (A000584).

Original entry on oeis.org

1, 36, 381, 2336, 10326, 36552, 110022, 292512, 704847, 1567852, 3263403, 6422208, 12046268, 21675408, 37608828, 63194304, 103199469, 164281524, 255573769, 389409504, 582206130, 855534680, 1237402530, 1763779680, 2480401755, 3444885756, 4729197591, 6422513536, 8634521016, 11499207456

Offset: 1

Luciano Ancora, Feb 05 2015

			Fifth differences:   1, 27,  93,  119,   120, (repeat 120) (A101100)
Fourth differences:  1, 28, 121,  240,   360,   480, ...   (A101095)
Third differences:   1, 29, 150,  390,   750,  1230, ...   (A101096)
Second differences:  1, 30, 180,  570,  1320,  2550, ...   (A101098)
First differences:   1, 31, 211,  781,  2101,  4651, ...   (A022521)
-------------------------------------------------------------------------
The fifth powers:    1, 32, 243, 1024,  3125,  7776, ...   (A000584)
-------------------------------------------------------------------------
First partial sums:  1, 33, 276, 1300,  4425, 12201, ...   (A000539)
Second partial sums: 1, 34, 310, 1610,  6035, 18236, ...   (A101092)
Third partial sums:  1, 35, 345, 1955,  7990, 26226, ...   (A101099)
Fourth partial sums: 1, 36, 381, 2336, 10326, 36552, ...   (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(10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A000539, A000584, A022521, A101092, A101095, A101096, A101098, A101099, A101100.

Cf. A101091 (fourth partial sums of fourth powers).

List([1..30], n-> Binomial(n+4,5)*(5*(n+2)^4 -35*(n+2)^2 +36)/126); # G. C. Greubel, Aug 28 2019

[Binomial(n+4,5)*(5*(n+2)^4 -35*(n+2)^2 +36)/126: n in [1..30]]; // G. C. Greubel, Aug 28 2019

seq(binomial(n+4,5)*(5*(n+2)^4 -35*(n+2)^2 +36)/126, n=1..30); # G. C. Greubel, Aug 28 2019

Table[n(1+n)(2+n)(3+n)(4+n)(-24 +20n +85n^2 +40n^3 +5n^4)/15120, {n, 30}] (* or *) Accumulate[Accumulate[Accumulate[Accumulate[Range[24]^5]]]] (* or *) CoefficientList[Series[(1 +26x +66x^2 +26x^3 +x^4)/(1-x)^10, {x, 0, 30}], x]
Nest[Accumulate,Range[30]^5,4] (* or *) LinearRecurrence[{10,-45,120, -210,252,-210,120,-45,10,-1}, {1,36,381,2336,10326,36552,110022,292512, 704847,1567852},30] (* Harvey P. Dale, May 08 2016 *)

vector(30, n, m=n+2; binomial(m+2,5)*(5*m^4 -35*m^2 +36)/126) \\ G. C. Greubel, Aug 28 2019

[binomial(n+4,5)*(5*(n+2)^4 -35*(n+2)^2 +36)/126 for n in (1..30)] # G. C. Greubel, Aug 28 2019

G.f.: x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1 - x)^10.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(-24 + 20*n + 85*n^2 + 40*n^3 + 5*n^4)/15120.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + n^5.

Edited by Bruno Berselli, Feb 10 2015

A254682 Fifth partial sums of fifth powers (A000584).

Original entry on oeis.org

1, 37, 418, 2754, 13080, 49632, 159654, 452166, 1157013, 2724865, 5988268, 12410476, 24456744, 46132152, 83740980, 146935284, 250134753, 414416277, 669990046, 1059399550, 1641605680, 2497140360, 3734542890, 5498322570

Offset: 1

Luciano Ancora, Feb 12 2015

			Fifth differences:   1, 27,  93,  119,   120, (repeat 120) (A101100)
Fourth differences:  1, 28, 121,  240,   360,   480, ...   (A101095)
Third differences:   1, 29, 150,  390,   750,  1230, ...   (A101096)
Second differences:  1, 30, 180,  570,  1320,  2550, ...   (A101098)
First differences:   1, 31, 211,  781,  2101,  4651, ...   (A022521)
-------------------------------------------------------------------------
The fifth powers:    1, 32, 243, 1024,  3125,  7776, ...   (A000584)
-------------------------------------------------------------------------
First partial sums:  1, 33, 276, 1300,  4425, 12201, ...   (A000539)
Second partial sums: 1, 34, 310, 1610,  6035, 18236, ...   (A101092)
Third partial sums:  1, 35, 345, 1955,  7990, 26226, ...   (A101099)
Fourth partial sums: 1, 36, 381, 2336, 10326, 36552, ...   (A254644)
Fifth partial sums:  1, 37, 418, 2754, 13080, 49632, ...   (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 (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A000539, A000584, A022521, A101092, A101095, A101096, A101098, A101099, A101100, A254644, A254681, A254683, A254684.

Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (- 2 + 5 n + n^2) (9 + 10 n + 2 n^2)/60480, {n,24}] (* or *)
CoefficientList[Series[(- 1 - 26 x - 66 x^2 - 26 x^3 - x^4)/(- 1 + x)^11, {x,0,23}], x]
Nest[Accumulate,Range[30]^5,5] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,37,418,2754,13080,49632,159654,452166,1157013,2724865,5988268},30] (* Harvey P. Dale, Jan 30 2019 *)

a(n)=n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(-2+5*n+n^2)*(9+10*n+2*n^2)/60480 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (- x - 26*x^2 - 66*x^3 - 26*x^4 - x^5)/(- 1 + x)^11.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(- 2 + 5*n + n^2)*(9 + 10*n + 2*n^2)/60480.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + n^5.
Sum_{n>=1} 1/a(n) = 475867/180 - (2560/13)*sqrt(7)*Pi*tan(sqrt(7)*Pi/2) + (210/13)*sqrt(3/11)*Pi*tan(sqrt(33)*Pi/2). - Amiram Eldar, Jan 27 2022

A101095 Fourth difference of fifth powers (A000584).

Original entry on oeis.org

1, 28, 121, 240, 360, 480, 600, 720, 840, 960, 1080, 1200, 1320, 1440, 1560, 1680, 1800, 1920, 2040, 2160, 2280, 2400, 2520, 2640, 2760, 2880, 3000, 3120, 3240, 3360, 3480, 3600, 3720, 3840, 3960, 4080, 4200, 4320, 4440, 4560, 4680, 4800, 4920, 5040, 5160, 5280

Offset: 1

Cecilia Rossiter, Dec 15 2004

Original Name: Shells (nexus numbers) of shells of shells of shells of the power of 5.

The (Worpitzky/Euler/Pascal Cube) "MagicNKZ" algorithm is: MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n + 1 - z, j)*(k - j + 1)^n, with k>=0, n>=1, z>=0. MagicNKZ is used to generate the n-th accumulation sequence of the z-th row of the Euler Triangle (A008292). For example, MagicNKZ(3,k,0) is the 3rd row of the Euler Triangle (followed by zeros) and MagicNKZ(10,k,1) is the partial sums of the 10th row of the Euler Triangle. This sequence is MagicNKZ(5,k-1,2).

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
D. J. Pengelley, The bridge between the continuous and the discrete via original sources in Study the Masters: The Abel-Fauvel Conference [pdf], Kristiansand, 2002, (ed. Otto Bekken et al), National Center for Mathematics Education, University of Gothenburg, Sweden, in press.
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Archive Machine link]
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013]
Eric Weisstein, Link to section of MathWorld: Worpitzky's Identity of 1883
Eric Weisstein, Link to section of MathWorld: Eulerian Number
Eric Weisstein, Link to section of MathWorld: Nexus number
Eric Weisstein, Link to section of MathWorld: Finite Differences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Fourth differences of A000584, third differences of A022521, second differences of A101098, and first differences of A101096.
For other sequences based upon MagicNKZ(n,k,z):
...... |  n = 1  |  n = 2  |  n = 3  |  n = 4  |  n = 5  |  n = 6  |  n = 7  |  n = 8
--------------------------------------------------------------------------------------
z =  0 | A000007 | A019590 | .......  MagicNKZ(n,k,0) = T(n,k+1) from A008292  .......
z =  1 | A000012 | A040000 | A101101 | A101104 | A101100 | ....... | ....... | .......
z =  2 | A000027 | A005408 | A008458 | A101103 | thisSeq | ....... | ....... | .......
z =  3 | A000217 | A000290 | A003215 | A005914 | A101096 | ....... | ....... | .......
z =  4 | A000292 | A000330 | A000578 | A005917 | A101098 | ....... | ....... | .......
z =  5 | A000332 | A002415 | A000537 | A000583 | A022521 | ....... | A255181 | .......
z =  6 | A000389 | A005585 | A024166 | A000538 | A000584 | A022522 | A255177 | A255182
z =  7 | A000579 | A040977 | A101094 | A101089 | A000539 | A001014 | A022523 | A255178
z =  8 | A000580 | A050486 | A101097 | A101090 | A101092 | A000540 | A001015 | A022524
z =  9 | A000581 | A053347 | A101102 | A101091 | A101099 | A101093 | A000541 | A001016
z = 10 | A000582 | A054333 | A254469 | A254681 | A254644 | A254640 | A250212 | A000542
z = 11 | A001287 | A054334 | A254869 | A254470 | A254682 | A254645 | A254641 | A253636
z = 12 | A001288 | A057788 | ....... | A254870 | A254471 | A254683 | A254646 | A254642
z = 13 | A010965 | ....... | ....... | ....... | A254871 | A254472 | A254684 | A254647
z = 14 | A010966 | ....... | ....... | ....... | ....... | A254872 | ....... | .......
--------------------------------------------------------------------------------------
Cf. A047969.

I:=[1,28,121,240,360]; [n le 5 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, May 07 2015

MagicNKZ=Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}];Table[MagicNKZ, {n, 5, 5}, {z, 2, 2}, {k, 0, 34}]
CoefficientList[Series[(1 + 26 x + 66 x^2 + 26 x^3 + x^4)/(1 - x)^2, {x, 0, 50}], x] (* Vincenzo Librandi, May 07 2015 *)
Join[{1,28,121,240},Differences[Range[50]^5,4]] (* or *) LinearRecurrence[{2,-1},{1,28,121,240,360},50] (* Harvey P. Dale, Jun 11 2016 *)

a(n)=if(n>3, 120*n-240, 33*n^2-72*n+40) \\ Charles R Greathouse IV, Oct 11 2015

[1,28,121]+[120*(k-2) for k in range(4,36)] # Danny Rorabaugh, Apr 23 2015

a(k+1) = Sum_{j=0..k+1} (-1)^j*binomial(n + 1 - z, j)*(k - j + 1)^n; n = 5, z = 2.
For k>3, a(k) = Sum_{j=0..4} (-1)^j*binomial(4, j)*(k - j)^5 = 120*(k - 2).
a(n) = 2*a(n-1) - a(n-2), n>5. G.f.: x*(1+26*x+66*x^2+26*x^3+x^4) / (1-x)^2. - Colin Barker, Mar 01 2012

MagicNKZ material edited, Crossrefs table added, SeriesAtLevelR material removed by Danny Rorabaugh, Apr 23 2015

Name changed and keyword 'uned' removed by Danny Rorabaugh, May 06 2015

A254645 Fourth partial sums of sixth powers (A001014).

Original entry on oeis.org

1, 68, 995, 7672, 40614, 166992, 571626, 1701480, 4534959, 11050468, 24997973, 53113424, 106959580, 205628736, 379603812, 676144944, 1166649837, 1956528420, 3198236503, 5108229896, 7988730530, 12255340240

Offset: 1

Luciano Ancora, Feb 05 2015

			First differences:   1, 63, 665, 3367, 11529,  31031, ...  (A022522)
--------------------------------------------------------------------------
The sixth powers:    1, 64, 729, 4096, 15625,  46656, ...  (A001014)
--------------------------------------------------------------------------
First partial sums:  1, 65, 794, 4890, 20515,  67171, ...  (A000540)
Second partial sums: 1, 66, 860, 5750, 26265,  93436, ...  (A101093)
Third partial sums:  1, 67, 927, 6677, 32942, 126378, ...  (A101099)
Fourth partial sums: 1, 68, 995, 7672, 40614, 166992, ...  (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 (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A000540, A001014, A022522, A101093, A101099.

Cf. A254644 (fourth partial sums of fifth powers), A254646 (fourth partial sums of seventh powers).

List([1..30], n-> Binomial(n+4,5)*(n+2)*((n^2+4*n-1)^2-2)/42); # G. C. Greubel, Aug 28 2019

[Binomial(n+4,5)*(n+2)*((n^2+4*n-1)^2-2)/42: n in [1..30]]; // G. C. Greubel, Aug 28 2019

seq(binomial(n+4,5)*(n+2)*((n^2+4*n-1)^2-2)/42, n=1..30); # G. C. Greubel, Aug 28 2019

Table[n (1 + n) (2 + n)^2 (3 + n) (4 + n) (- 1 - 8 n + 14 n^2 + 8 n^3 + n^4)/5040, {n, 22}] (* or *)
Accumulate[Accumulate[Accumulate[Accumulate[Range[22]^6]]]] (* or *)
CoefficientList[Series[(- 1 - 57 x - 302 x^2 - 302 x^3 - 57 x^4 - x^5)/(- 1 + x)^11, {x, 0, 21}], x]
Nest[Accumulate,Range[30]^6,4] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,68,995,7672,40614,166992,571626,1701480,4534959,11050468,24997973},30] (* Harvey P. Dale, Dec 27 2015 *)

vector(30, n, binomial(n+4,5)*(n+2)*((n^2+4*n-1)^2-2)/42) \\ G. C. Greubel, Aug 28 2019

[binomial(n+4,5)*(n+2)*((n^2+4*n-1)^2-2)/42 for n in (1..30)] # G. C. Greubel, Aug 28 2019

G.f.: x*(1 + 57*x + 302*x^2 + 302*x^3 + 57*x^4 + x^5)/(1 - x)^11.
a(n) = n*(1 + n)*(2 + n)^2*(3 + n)*(4 + n)*(- 1 - 8*n + 14*n^2 + 8*n^3 + n^4)/5040.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + n^6.

A254683 Fifth partial sums of sixth powers (A001014).

Original entry on oeis.org

1, 69, 1064, 8736, 49350, 216342, 787968, 2489448, 7024407, 18074875, 43072848, 96186272, 203145852, 408774588, 788378400, 1464523344, 2631173181, 4587701601, 7785938104, 12894168000, 20882898530, 33138238770

Offset: 1

Luciano Ancora, Feb 12 2015

			First differences:   1, 63,  665, 3367, 11529, ...  (A022522)
--------------------------------------------------------------------------
The sixth powers:    1, 64,  729, 4096, 15625, ...  (A001014)
--------------------------------------------------------------------------
First partial sums:  1, 65,  794, 4890, 20515, ...  (A000540)
Second partial sums: 1, 66,  860, 5750, 26265, ...  (A101093)
Third partial sums:  1, 67,  927, 6677, 32942, ...  (A254640)
Fourth partial sums: 1, 68,  995, 7672, 40614, ...  (A254645)
Fifth partial sums:  1, 69, 1064, 8736, 49350, ...  (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 (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A000540, A001014, A022522, A101093, A254640, A254645, A254681, A254682, A254684.

Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (5 + 2*n) (- 3 + 5*n + n^2) (4 + 15 n + 3 n^2)/332640, {n,22}] (* or *)
CoefficientList[Series[(1 + 57 x + 302 x^2 + 302 x^3 + 57 x^4 + x^5)/(- 1 + x)^12, {x,0,21}], x]

G.f.: (x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)/(- 1 + x)^12.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(5 + 2*n)*(- 3 + 5*n + n^2)*(4 + 15*n + 3*n^2)/332640.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + n^6.

cp's OEIS Frontend

A000538 Sum of fourth powers: 0^4 + 1^4 + ... + n^4.

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A254647 Fourth partial sums of eighth powers (A001016).

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A101092 Second partial sums of fifth powers (A000584).

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A254681 Fifth partial sums of fourth powers (A000583).

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A254469 Sixth partial sums of cubes (A000578).

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A254644 Fourth partial sums of fifth powers (A000584).