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A101096 Third differences of fifth powers (A000584).

%I A101096 #60 Mar 18 2025 16:38:03
%S A101096 1,29,150,390,750,1230,1830,2550,3390,4350,5430,6630,7950,9390,10950,
%T A101096 12630,14430,16350,18390,20550,22830,25230,27750,30390,33150,36030,
%U A101096 39030,42150,45390,48750,52230,55830,59550,63390,67350,71430,75630,79950,84390,88950
%N A101096 Third differences of fifth powers (A000584).
%C A101096 Original Name: Shells (nexus numbers) of shells of shells of the power of 5.
%C A101096 For n>=3 a(n) is equal to the number of functions f:{1,2,3,4,5}->{1,2,...,n} such that Im(f) contains 3 fixed elements. - Aleksandar M. Janjic and _Milan Janjic_, Feb 24 2007
%H A101096 Danny Rorabaugh, <a href="/A101096/b101096.txt">Table of n, a(n) for n = 1..10000</a>
%H A101096 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>.
%H A101096 David J. Pengelley, <a href="http://sofia.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, 2003.
%H A101096 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 A101096 Cecilia Rossiter, <a href="/A101096/a101096.pdf">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a> [Cached copy, May 15 2013]
%H A101096 Eric Weisstein, Link to section of MathWorld: <a href="https://mathworld.wolfram.com/WorpitzkysIdentity.html">Worpitzky's Identity of 1883</a>.
%H A101096 Eric Weisstein, Link to section of MathWorld: <a href="https://mathworld.wolfram.com/EulerianNumber.html">Eulerian Number</a>.
%H A101096 Eric Weisstein, Link to section of MathWorld: <a href="https://mathworld.wolfram.com/NexusNumber.html">Nexus number</a>.
%H A101096 Eric Weisstein, Link to section of MathWorld: <a href="https://mathworld.wolfram.com/FiniteDifference.html">Finite Differences</a>.
%H A101096 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A101096 a(k+1) = MagicNKZ(5,k,3) where MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n+1-z,j)*(k-j+1)^n. (Cf. A101095.)
%F A101096 a(n+1) = 30*(1 - 2*n + 2*n^2) for n>2.
%F A101096 a(n+3) = A069477(n). - _Vladimir Joseph Stephan Orlovsky_, Jun 19 2011
%F A101096 G.f.: x*(x^4+26*x^3+66*x^2+26*x+1)/(1-x)^3. - _Colin Barker_, Oct 17 2012
%F A101096 Sum_{n>=1} 1/a(n) = (Pi/60)*tanh(Pi/2) + 871/870. - _Amiram Eldar_, Jan 27 2022
%t A101096 MagicNKZ=Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}];Table[MagicNKZ, {n, 5, 5}, {z, 3, 3}, {k, 0, 34}]
%t A101096 CoefficientList[Series[(-z^4-26z^3-66z^2-26z-1)/(z-1)^3, {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 19 2011 *)
%t A101096 Join[{1,29},Differences[Range[0,40]^5,3]] (* or *) LinearRecurrence[{3,-3,1},{1,29,150,390,750},40] (* _Harvey P. Dale_, Feb 02 2017 *)
%o A101096 (Sage) [sum([(-1)^j*binomial(3, j)*(k-j+1)^5 for j in range(min(k+2,4))]) for k in range(40)] # _Danny Rorabaugh_, Apr 27 2015
%o A101096 (PARI) a(n)=if(n>2,60*n^2-180*n+150,28*n-27) \\ _Charles R Greathouse IV_, Oct 11 2015
%o A101096 (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( x*(x^4+26*x^3+66*x^2+26*x+1)/(1-x)^3)); // _G. C. Greubel_, Dec 01 2018
%Y A101096 Cf. A069477.
%Y A101096 Third differences of A000584, second differences of A022521, and first differences of A068236.
%Y A101096 Cf. A101095 for other sequences related to MagicNKZ.
%Y A101096 Cf. A001844.
%K A101096 easy,nonn
%O A101096 1,2
%A A101096 Cecilia Rossiter, Dec 15 2004
%E A101096 MagicNKZ material edited and SeriesAtLevelR material removed by _Danny Rorabaugh_, Apr 27 2015