4,3

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 seen as:

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]

t(n,m)=(Eulerian[n,m]-Binomial[n,m]^2)/2:

starting at n=4 and m={2,n-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},

{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}

Clear[p, x, n]; 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[%]

Sequence in context: A082269 A107776 A161202 this_sequence A114332 A077348 A113259

Adjacent sequences: A154350 A154351 A154352 this_sequence A154354 A154355 A154356

nonn,uned,tabf

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 07 2009