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a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4.
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%I #41 Sep 23 2015 15:28:37

%S 1,12,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,

%T 24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,

%U 24,24,24,24,24,24,24,24,24,24,24,24,24

%N a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4.

%C Original name: The first summation of row 4 of Euler's triangle - a row that will recursively accumulate to the power of 4.

%H 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 C. Rossiter, <a href="http://noticingnumbers.net/300SeriesCube.htm">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a> [Dead link]

%H C. Rossiter, <a href="/A101104/a101104.pdf">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a> [Cached copy, May 15 2013]

%H Eric Weisstein, Link to section of MathWorld: <a href="http://mathworld.wolfram.com/WorpitzkysIdentity.html">Worpitzky's Identity of 1883</a>

%H Eric Weisstein, Link to section of MathWorld: <a href="http://mathworld.wolfram.com/EulerianNumber.html">Eulerian Number</a>

%H Eric Weisstein, Link to section of MathWorld: <a href="http://mathworld.wolfram.com/NexusNumber.html">Nexus number</a>

%H Eric Weisstein, Link to section of MathWorld: <a href="http://mathworld.wolfram.com/FiniteDifference.html">Finite Differences</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).

%F a(k) = MagicNKZ(4,k,1) where MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n+1-z,j)*(k-j+1)^n (cf. A101095). That is, a(k) = Sum_{j=0..k+1} (-1)^j*binomial(4, j)*(k-j+1)^4.

%F a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4. - _Joerg Arndt_, Nov 30 2014

%F G.f.: x*(1+11*x+11*x^2+x^3)/(1-x). - _Colin Barker_, Apr 16 2012

%t MagicNKZ = Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}];Table[MagicNKZ, {n, 4, 4}, {z, 1, 1}, {k, 0, 34}]

%t Join[{1, 12, 23},LinearRecurrence[{1},{24},56]] (* _Ray Chandler_, Sep 23 2015 *)

%Y For other sequences based upon MagicNKZ(n,k,z):

%Y ..... | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 | n = 6 | n = 7

%Y ---------------------------------------------------------------------------

%Y z = 0 | A000007 | A019590 | .......MagicNKZ(n,k,0) = A008292(n,k+1) .......

%Y z = 1 | A000012 | A040000 | A101101 | thisSeq | A101100 | ....... | .......

%Y z = 2 | A000027 | A005408 | A008458 | A101103 | A101095 | ....... | .......

%Y z = 3 | A000217 | A000290 | A003215 | A005914 | A101096 | ....... | .......

%Y z = 4 | A000292 | A000330 | A000578 | A005917 | A101098 | ....... | .......

%Y z = 5 | A000332 | A002415 | A000537 | A000583 | A022521 | ....... | A255181

%Y z = 6 | A000389 | A005585 | A024166 | A000538 | A000584 | A022522 | A255177

%Y z = 7 | A000579 | A040977 | A101094 | A101089 | A000539 | A001014 | A022523

%Y z = 8 | A000580 | A050486 | A101097 | A101090 | A101092 | A000540 | A001015

%Y z = 9 | A000581 | A053347 | A101102 | A101091 | A101099 | A101093 | A000541

%Y Cf. A101095 for an expanded table and more about MagicNKZ.

%K easy,nonn

%O 1,2

%A Cecilia Rossiter, Dec 15 2004

%E New name from _Joerg Arndt_, Nov 30 2014

%E Original Formula edited and Crossrefs table added by _Danny Rorabaugh_, Apr 22 2015