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A101104 a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4.

%I A101104 #42 Feb 16 2025 08:32:55
%S A101104 1,12,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,
%T A101104 24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,
%U A101104 24,24,24,24,24,24,24,24,24,24,24,24,24
%N A101104 a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4.
%C A101104 Original name: The first summation of row 4 of Euler's triangle - a row that will recursively accumulate to the power of 4.
%H A101104 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 A101104 C. Rossiter, <a href="http://noticingnumbers.net/300SeriesCube.htm">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a> [Dead link]
%H A101104 C. Rossiter, <a href="/A101104/a101104.pdf">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a> [Cached copy, May 15 2013]
%H A101104 Eric Weisstein, Link to section of MathWorld: <a href="https://mathworld.wolfram.com/WorpitzkysIdentity.html">Worpitzky's Identity of 1883</a>
%H A101104 Eric Weisstein, Link to section of MathWorld: <a href="https://mathworld.wolfram.com/EulerianNumber.html">Eulerian Number</a>
%H A101104 Eric Weisstein, Link to section of MathWorld: <a href="https://mathworld.wolfram.com/NexusNumber.html">Nexus number</a>
%H A101104 Eric Weisstein, Link to section of MathWorld: <a href="https://mathworld.wolfram.com/FiniteDifference.html">Finite Differences</a>
%H A101104 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F A101104 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 A101104 a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4. - _Joerg Arndt_, Nov 30 2014
%F A101104 G.f.: x*(1+11*x+11*x^2+x^3)/(1-x). - _Colin Barker_, Apr 16 2012
%t A101104 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 A101104 Join[{1, 12, 23},LinearRecurrence[{1},{24},56]] (* _Ray Chandler_, Sep 23 2015 *)
%Y A101104 For other sequences based upon MagicNKZ(n,k,z):
%Y A101104 ..... |  n = 1  |  n = 2  |  n = 3  |  n = 4  |  n = 5  |  n = 6  |  n = 7
%Y A101104 ---------------------------------------------------------------------------
%Y A101104 z = 0 | A000007 | A019590 | .......MagicNKZ(n,k,0) = A008292(n,k+1) .......
%Y A101104 z = 1 | A000012 | A040000 | A101101 | thisSeq | A101100 | ....... | .......
%Y A101104 z = 2 | A000027 | A005408 | A008458 | A101103 | A101095 | ....... | .......
%Y A101104 z = 3 | A000217 | A000290 | A003215 | A005914 | A101096 | ....... | .......
%Y A101104 z = 4 | A000292 | A000330 | A000578 | A005917 | A101098 | ....... | .......
%Y A101104 z = 5 | A000332 | A002415 | A000537 | A000583 | A022521 | ....... | A255181
%Y A101104 z = 6 | A000389 | A005585 | A024166 | A000538 | A000584 | A022522 | A255177
%Y A101104 z = 7 | A000579 | A040977 | A101094 | A101089 | A000539 | A001014 | A022523
%Y A101104 z = 8 | A000580 | A050486 | A101097 | A101090 | A101092 | A000540 | A001015
%Y A101104 z = 9 | A000581 | A053347 | A101102 | A101091 | A101099 | A101093 | A000541
%Y A101104 Cf. A101095 for an expanded table and more about MagicNKZ.
%K A101104 easy,nonn
%O A101104 1,2
%A A101104 Cecilia Rossiter, Dec 15 2004
%E A101104 New name from _Joerg Arndt_, Nov 30 2014
%E A101104 Original Formula edited and Crossrefs table added by _Danny Rorabaugh_, Apr 22 2015