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A174035 A triangle sequence of the form: T(n,m) = binomial(n, m) + floor(Eulerian(n + 1, m)/2).

%I A174035 #5 May 10 2025 23:16:00
%S A174035 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,
%T A174035 66,1,1,130,2167,7844,7844,2167,130,1,1,259,7332,44173,78165,44173,
%U A174035 7332,259,1,1,515,23956,227680,655303,655303,227680,23956,515,1,1,1028,76363,1101864,4869267,7862376,4869267,1101864,76363,1028,1
%N A174035 A triangle sequence of the form: T(n,m) = binomial(n, m) + floor(Eulerian(n + 1, m)/2).
%C A174035 Row sums are: {1, 2, 6, 18, 75, 390, 2582, 20284, 181695, 1814910, 19959422,...}.
%F A174035 T(n,m) = binomial(n, m) + floor(Eulerian(n + 1, m)/2).
%F A174035 T(n,m) = A007318(n, m) + floor(A008292(n + 1, m + 1)/2). - _Jason Yuen_, May 10 2025
%e A174035 Triangle begins:
%e A174035   {1},
%e A174035   {1, 1},
%e A174035   {1, 4, 1},
%e A174035   {1, 8, 8, 1},
%e A174035   {1, 17, 39, 17, 1},
%e A174035   {1, 33, 161, 161, 33, 1},
%e A174035   {1, 66, 610, 1228, 610, 66, 1},
%e A174035   {1, 130, 2167, 7844, 7844, 2167, 130, 1},
%e A174035   {1, 259, 7332, 44173, 78165, 44173, 7332, 259, 1},
%e A174035   {1, 515, 23956, 227680, 655303, 655303, 227680, 23956, 515, 1},
%e A174035   {1, 1028, 76363, 1101864, 4869267, 7862376, 4869267, 1101864, 76363, 1028, 1}
%t A174035 << DiscreteMath`Combinatorica`
%t A174035 Table[Table[Binomial[n, m] + Floor[Eulerian[n + 1, m] 2], {m, 0, n}], {n, 0, 10}];
%t A174035 Flatten[%]
%Y A174035 Cf. A007318, A008292, A174098.
%K A174035 nonn,tabl,uned
%O A174035 0,5
%A A174035 _Roger L. Bagula_, Mar 06 2010