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A101100 The first summation of row 5 of Euler's triangle - a row that will recursively accumulate to the power of 5.

%I A101100 #28 Feb 16 2025 08:32:55
%S A101100 1,27,93,119,120,120,120,120,120,120,120,120,120,120,120,120,120,120,
%T A101100 120,120,120,120,120,120,120,120,120,120,120,120,120,120,120,120,120
%N A101100 The first summation of row 5 of Euler's triangle - a row that will recursively accumulate to the power of 5.
%D A101100 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 533.
%H A101100 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 A101100 C. Rossiter, <a href="http://noticingnumbers.net/300SeriesCube.htm">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a>.
%H A101100 Eric Weisstein's World of Mathematics <a href="https://mathworld.wolfram.com/WorpitzkysIdentity.html">Worpitzky's Identity of 1883</a>.
%H A101100 Eric Weisstein's World of Mathematics <a href="https://mathworld.wolfram.com/EulerianNumber.html">Eulerian Number</a>.
%H A101100 Eric Weisstein's World of Mathematics <a href="https://mathworld.wolfram.com/NexusNumber.html">Nexus number</a>.
%H A101100 Eric Weisstein's World of Mathematics <a href="https://mathworld.wolfram.com/FiniteDifference.html">Finite Differences</a>.
%H A101100 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F A101100 a(n) = 120, n>4.
%F A101100 a(n) = Sum_{j=1..m} Eulerian(m, j-1)*binomial(m+n-j+r, m+r), with m = 5, r = -5.
%F A101100 a(n) = Sum_{j=0..n+1} (-1)^j*binomial(m+1-z, j)*(n-j+1)^n, with m = 5, z = 1.
%F A101100 G.f.: x*(1+26*x+66*x^2+26*x^3+x^4)/(1-x). - _Colin Barker_, Mar 01 2012
%t A101100 MagicNKZ=Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}]; Table[MagicNKZ, {n, 5, 5}, {z, 1, 1}, {k, 0, 34}]
%t A101100 (* or *)
%t A101100 SeriesAtLevelR = Sum[Eulerian[n, i-1]*Binomial[n+x-i+r, n+r], {i,1,n}]; Table[SeriesAtLevelR, {n, 5, 5}, {r, -5, -5}, {x, 5, 35}]
%o A101100 (PARI) {a(n) = if(n==1, 1, if(n==2, 27, if(n==3, 93, if(n==4, 119, 120))) )}; \\ _G. C. Greubel_, May 07 2019
%o A101100 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x*(1+26*x+66*x^2+26*x^3+x^4)/(1-x) )); // _G. C. Greubel_, May 07 2019
%o A101100 (Sage) a=(x*(1+26*x+66*x^2+26*x^3+x^4)/(1-x)).series(x, 40).coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, May 07 2019
%Y A101100 Within the "cube" of related sequences with construction based upon MaginNKZ formula, with n downward, k rightward and z backward: Before: this sequence, A101095, A101096, A101098, A022521, A000584, A000539, A101092, A101099. Above: A101104, this sequence.
%Y A101100 Within the "cube" of related sequences with construction based upon SeriesAtLevelR formula, with n downward, x rightward and r backward: Before: this sequence, A101095, A101096, A101098, A022521, A000584, A000539, A101092, A101099.
%K A101100 easy,nonn
%O A101100 1,2
%A A101100 Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004