login
A154353
Triangle T(n,m) read by rows: T(n,m) = ( Eulerian(n,m) - Binomial(n,m)^2 )/2, n >= 4, 2 <= m = <= n-1.
1
1, 1, 5, 15, 5, 16, 101, 101, 16, 42, 483, 1008, 483, 42, 99, 1926, 7197, 7197, 1926, 99, 219, 6912, 42549, 75645, 42549, 6912, 219, 466, 23272, 224068, 647239, 647239, 224068, 23272, 466, 968, 75306, 1094544, 4847007, 7830372, 4847007, 1094544, 75306, 968
OFFSET
4,3
COMMENTS
Row sums are: {2, 25, 234, 2058, 18444, 175005, 1790090, 19866022, 239148084, 3112158322, 43583945300,...}.
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.
The resulting fractal can be obtained from the Mathematica code given in the Mathematica code section.
FORMULA
T(n,m) = ( Eulerian(n,m) - Binomial(n,m)^2 )/2 for n >= 4, and 2 <= m <= n-1.
EXAMPLE
{1, 1},
{5, 15, 5},
{16, 101, 101, 16},
{42, 483, 1008, 483, 42},
{99, 1926, 7197, 7197, 1926, 99},
{219, 6912, 42549, 75645, 42549, 6912, 219},
{466, 23272, 224068, 647239, 647239, 224068, 23272, 466},
{968, 75306, 1094544, 4847007, 7830372, 4847007, 1094544, 75306, 968},
{1981, 237623, 5080230, 33104787, 81149421, 81149421, 33104787, 5080230, 237623, 1981},
{4017, 737685, 22742525, 211518255, 752497122, 1137159114, 752497122, 211518255, 22742525, 737685, 4017},
{8100, 2265615, 99164495, 1285615475, 6420803247, 13984115718, 13984115718, 6420803247, 1285615475, 99164495, 2265615, 8100}
MATHEMATICA
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[%]
(* fractal code *)
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}];
b = Table[If[m <= n + 1, Mod[a[[n]][[m]], 2], 0], {m, 1, Length[a]}, {n, 1, Length[a]}]; ListDensityPlot[b, Mesh -> False]
CROSSREFS
Sequence in context: A107776 A290528 A161202 * A077348 A290829 A290837
KEYWORD
nonn,tabf
AUTHOR
Roger L. Bagula, Jan 07 2009
STATUS
approved