A174035 A triangle sequence of the form: T(n,m) = binomial(n, m) + floor(Eulerian(n + 1, m)/2).
1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 17, 39, 17, 1, 1, 33, 161, 161, 33, 1, 1, 66, 610, 1228, 610, 66, 1, 1, 130, 2167, 7844, 7844, 2167, 130, 1, 1, 259, 7332, 44173, 78165, 44173, 7332, 259, 1, 1, 515, 23956, 227680, 655303, 655303, 227680, 23956, 515, 1, 1, 1028, 76363, 1101864, 4869267, 7862376, 4869267, 1101864, 76363, 1028, 1
Offset: 0
Examples
Triangle begins:
{1},
{1, 1},
{1, 4, 1},
{1, 8, 8, 1},
{1, 17, 39, 17, 1},
{1, 33, 161, 161, 33, 1},
{1, 66, 610, 1228, 610, 66, 1},
{1, 130, 2167, 7844, 7844, 2167, 130, 1},
{1, 259, 7332, 44173, 78165, 44173, 7332, 259, 1},
{1, 515, 23956, 227680, 655303, 655303, 227680, 23956, 515, 1},
{1, 1028, 76363, 1101864, 4869267, 7862376, 4869267, 1101864, 76363, 1028, 1}
Programs
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Mathematica
<< DiscreteMath`Combinatorica` Table[Table[Binomial[n, m] + Floor[Eulerian[n + 1, m] 2], {m, 0, n}], {n, 0, 10}]; Flatten[%]
Formula
T(n,m) = binomial(n, m) + floor(Eulerian(n + 1, m)/2).
Comments