0,5

Row sums are below (n+1)!:

{1, 2, 6, 12, 28, 52, 112, 204, 402, 744, 1414,...}.

The sequence is a designed sequence to be near the Eulerian numbers using p partitions.

The remarkable thing about this sub- Eulerian numbers sequence

is that the sequence is a Sierpinski gasket modulo 2:

Clear[a, b];

a = Table[Table[ Binomial[n, m], {m, 0, n}] + (Table[t[n, m], {m, 0, n}] + Reverse[Table[t[n, m], {m, 0, n}]]), {n, 0, 64}];

b = Table[If[m <= n, Mod[a[[n]][[m]], 2], 0], {m, 1, Length[a]}, {n, 1, Length[a]}];

ListDensityPlot[b, Mesh -> False, Frame -> False]

t0(n,m)=1 + PartitionsP[n] - PartitionsP[m] - PartitionsP[n - m];

t(n,m)=Binomial[n,m]+t0(n,m)+Reverse[t0(n,m)).

{1},

{1, 1},

{1, 4, 1},

{1, 5, 5, 1},

{1, 8, 10, 8, 1},

{1, 9, 16, 16, 9, 1},

{1, 14, 25, 32, 25, 14, 1},

{1, 15, 35, 51, 51, 35, 15, 1},

{1, 22, 48, 82, 96, 82, 48, 22, 1},

{1, 25, 64, 118, 164, 164, 118, 64, 25, 1},

{1, 34, 83, 170, 264, 310, 264, 170, 83, 34, 1}

Clear[f];

t[n_, m_] = 1 + PartitionsP[n] - PartitionsP[m] - PartitionsP[n - m]; \! Table[(Table[t[n, m], {m, 0, n}] + Reverse[Table[t[n, m], {m, 0, n}]])/2, {n, 0, 10}];

Table[Table[Binomial[n, m], {m, 0, n}] + (Table[t[n, m], {m, 0, n}] + Reverse[Table[t[n, m], {m, 0, n}]]), {n, 0, 10}];

Flatten[%]

Sequence in context: A147566 A146770 A143334 this_sequence A136489 A166455 A131239

Adjacent sequences: A156047 A156048 A156049 this_sequence A156051 A156052 A156053

nonn,tabl,uned

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 02 2009