A101095 Fourth difference of fifth powers (A000584).

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

Author

Cecilia Rossiter, Dec 15 2004

Keywords

• easy
• nonn

Comments

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).

Links

• 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).

Crossrefs

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

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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.

Programs

Magma

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

Mathematica

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 *)

PARI

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

Sage

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

Formula

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

Extensions

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