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A101095 Fourth difference of fifth powers (A000584). 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Dead link]

C. 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).

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

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 *)

PROG

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

(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

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

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

--------------------------------------------------------------------------------------

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.

Sequence in context: A233372 A220755 A244079 * A232403 A219851 A320885

Adjacent sequences:  A101092 A101093 A101094 * A101096 A101097 A101098

KEYWORD

easy,nonn

AUTHOR

Cecilia Rossiter, Dec 15 2004

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

STATUS

approved

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Last modified September 20 16:48 EDT 2019. Contains 327242 sequences. (Running on oeis4.)