|
%I #10 Sep 14 2016 08:35:54
%S 1,1,5,15,5,16,101,101,16,42,483,1008,483,42,99,1926,7197,7197,1926,
%T 99,219,6912,42549,75645,42549,6912,219,466,23272,224068,647239,
%U 647239,224068,23272,466,968,75306,1094544,4847007,7830372,4847007,1094544,75306,968
%N Triangle T(n,m) read by rows: T(n,m) = ( Eulerian(n,m) - Binomial(n,m)^2 )/2, n >= 4, 2 <= m = <= n-1.
%C Row sums are: {2, 25, 234, 2058, 18444, 175005, 1790090, 19866022, 239148084, 3112158322, 43583945300,...}.
%C Noticing the Eulerian numbers and the binomial squared were the same for the first four rows, I subtracted them and extracted the zeros to get this sequence.
%C The resulting fractal can be obtained from the Mathematica code given in the Mathematica code section.
%H G. C. Greubel, <a href="/A154353/b154353.txt">Table of n, a(n) for the first 50 rows</a>
%F T(n,m) = ( Eulerian(n,m) - Binomial(n,m)^2 )/2 for n >= 4, and 2 <= m <= n-1.
%e {1, 1},
%e {5, 15, 5},
%e {16, 101, 101, 16},
%e {42, 483, 1008, 483, 42},
%e {99, 1926, 7197, 7197, 1926, 99},
%e {219, 6912, 42549, 75645, 42549, 6912, 219},
%e {466, 23272, 224068, 647239, 647239, 224068, 23272, 466},
%e {968, 75306, 1094544, 4847007, 7830372, 4847007, 1094544, 75306, 968},
%e {1981, 237623, 5080230, 33104787, 81149421, 81149421, 33104787, 5080230, 237623, 1981},
%e {4017, 737685, 22742525, 211518255, 752497122, 1137159114, 752497122, 211518255, 22742525, 737685, 4017},
%e {8100, 2265615, 99164495, 1285615475, 6420803247, 13984115718, 13984115718, 6420803247, 1285615475, 99164495, 2265615, 8100}
%t p[x_, n_] = (x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x; Table[Table[(CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m]] - Binomial[n - 1, m - 1]^2)/2, {m, 2, n - 1}], {n, 4, 14}]; Flatten[%]
%t (* fractal code *)
%t a = Table[Table[(CoefficientList[FullSimplify[ExpandAll[p[x, n]]],x][[m]] - Binomial[n - 1,m - 1]^2)/2, {m, 2, n - 1}], {n, 4, 34}];
%t b = Table[If[m <= n + 1, Mod[a[[n]][[m]], 2], 0], {m, 1, Length[a]}, {n, 1, Length[a]}]; ListDensityPlot[b, Mesh -> False]
%K nonn,tabf
%O 4,3
%A _Roger L. Bagula_, Jan 07 2009
|
