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A101095 Fourth difference of fifth powers (A000584).

%I A101095 #54 Mar 11 2025 11:08:40
%S A101095 1,28,121,240,360,480,600,720,840,960,1080,1200,1320,1440,1560,1680,
%T A101095 1800,1920,2040,2160,2280,2400,2520,2640,2760,2880,3000,3120,3240,
%U A101095 3360,3480,3600,3720,3840,3960,4080,4200,4320,4440,4560,4680,4800,4920,5040,5160,5280
%N A101095 Fourth difference of fifth powers (A000584).
%C A101095 Original Name: Shells (nexus numbers) of shells of shells of shells of the power of 5.
%C A101095 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).
%H A101095 Danny Rorabaugh, <a href="/A101095/b101095.txt">Table of n, a(n) for n = 1..10000</a>
%H A101095 D. J. Pengelley, <a href="http://www.math.nmsu.edu/~davidp/bridge.pdf">The bridge between the continuous and the discrete via original sources in Study the Masters: The Abel-Fauvel Conference</a> [pdf], Kristiansand, 2002, (ed. Otto Bekken et al), National Center for Mathematics Education, University of Gothenburg, Sweden, in press.
%H A101095 Cecilia Rossiter, <a href="https://web.archive.org/web/20090428171006/http://noticingnumbers.net/300SeriesCube.htm">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a> [Archive Machine link]
%H A101095 Cecilia Rossiter, <a href="/A101095/a101095.pdf">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a> [Cached copy, May 15 2013]
%H A101095 Eric Weisstein, Link to section of MathWorld: <a href="https://mathworld.wolfram.com/WorpitzkysIdentity.html">Worpitzky's Identity of 1883</a>
%H A101095 Eric Weisstein, Link to section of MathWorld: <a href="https://mathworld.wolfram.com/EulerianNumber.html">Eulerian Number</a>
%H A101095 Eric Weisstein, Link to section of MathWorld: <a href="https://mathworld.wolfram.com/NexusNumber.html">Nexus number</a>
%H A101095 Eric Weisstein, Link to section of MathWorld: <a href="https://mathworld.wolfram.com/FiniteDifference.html">Finite Differences</a>
%H A101095 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A101095 a(k+1) = Sum_{j=0..k+1} (-1)^j*binomial(n + 1 - z, j)*(k - j + 1)^n; n = 5, z = 2.
%F A101095 For k>3, a(k) = Sum_{j=0..4} (-1)^j*binomial(4, j)*(k - j)^5 = 120*(k - 2).
%F A101095 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
%t A101095 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}]
%t A101095 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 *)
%t A101095 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 *)
%o A101095 (Sage) [1,28,121]+[120*(k-2) for k in range(4,36)] # _Danny Rorabaugh_, Apr 23 2015
%o A101095 (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
%o A101095 (PARI) a(n)=if(n>3, 120*n-240, 33*n^2-72*n+40) \\ _Charles R Greathouse IV_, Oct 11 2015
%Y A101095 Fourth differences of A000584, third differences of A022521, second differences of A101098, and first differences of A101096.
%Y A101095 For other sequences based upon MagicNKZ(n,k,z):
%Y A101095 ...... |  n = 1  |  n = 2  |  n = 3  |  n = 4  |  n = 5  |  n = 6  |  n = 7  |  n = 8
%Y A101095 --------------------------------------------------------------------------------------
%Y A101095 z =  0 | A000007 | A019590 | .......  MagicNKZ(n,k,0) = T(n,k+1) from A008292  .......
%Y A101095 z =  1 | A000012 | A040000 | A101101 | A101104 | A101100 | ....... | ....... | .......
%Y A101095 z =  2 | A000027 | A005408 | A008458 | A101103 | thisSeq | ....... | ....... | .......
%Y A101095 z =  3 | A000217 | A000290 | A003215 | A005914 | A101096 | ....... | ....... | .......
%Y A101095 z =  4 | A000292 | A000330 | A000578 | A005917 | A101098 | ....... | ....... | .......
%Y A101095 z =  5 | A000332 | A002415 | A000537 | A000583 | A022521 | ....... | A255181 | .......
%Y A101095 z =  6 | A000389 | A005585 | A024166 | A000538 | A000584 | A022522 | A255177 | A255182
%Y A101095 z =  7 | A000579 | A040977 | A101094 | A101089 | A000539 | A001014 | A022523 | A255178
%Y A101095 z =  8 | A000580 | A050486 | A101097 | A101090 | A101092 | A000540 | A001015 | A022524
%Y A101095 z =  9 | A000581 | A053347 | A101102 | A101091 | A101099 | A101093 | A000541 | A001016
%Y A101095 z = 10 | A000582 | A054333 | A254469 | A254681 | A254644 | A254640 | A250212 | A000542
%Y A101095 z = 11 | A001287 | A054334 | A254869 | A254470 | A254682 | A254645 | A254641 | A253636
%Y A101095 z = 12 | A001288 | A057788 | ....... | A254870 | A254471 | A254683 | A254646 | A254642
%Y A101095 z = 13 | A010965 | ....... | ....... | ....... | A254871 | A254472 | A254684 | A254647
%Y A101095 z = 14 | A010966 | ....... | ....... | ....... | ....... | A254872 | ....... | .......
%Y A101095 --------------------------------------------------------------------------------------
%Y A101095 Cf. A047969.
%K A101095 easy,nonn
%O A101095 1,2
%A A101095 Cecilia Rossiter, Dec 15 2004
%E A101095 MagicNKZ material edited, Crossrefs table added, SeriesAtLevelR material removed by _Danny Rorabaugh_, Apr 23 2015
%E A101095 Name changed and keyword 'uned' removed by _Danny Rorabaugh_, May 06 2015