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A141690 Triangle t(n,m) = 2*A008292(n+1,m+1) - A007318(n,m), a linear combination of Eulerian numbers and Pascal's triangle, 0 <= m <= n.

%I A141690 #17 Nov 26 2021 12:57:34
%S A141690 1,1,1,1,6,1,1,19,19,1,1,48,126,48,1,1,109,594,594,109,1,1,234,2367,
%T A141690 4812,2367,234,1,1,487,8565,31203,31203,8565,487,1,1,996,29188,176412,
%U A141690 312310,176412,29188,996,1,1,2017,95644,910300,2620582,2620582,910300
%N A141690 Triangle t(n,m) = 2*A008292(n+1,m+1) - A007318(n,m), a linear combination of Eulerian numbers and Pascal's triangle, 0 <= m <= n.
%C A141690 Row sums are 1, 2, 8, 40, 224, 1408, 10016, 80512, 725504, 7257088, ... = 2*(n+1)! - 2^n.
%H A141690 G. C. Greubel, <a href="/A141690/b141690.txt">Rows n = 0..100 of triangle, flattened</a>
%e A141690 Triangle begins
%e A141690   1;
%e A141690   1,    1;
%e A141690   1,    6,     1;
%e A141690   1,   19,    19,      1;
%e A141690   1,   48,   126,     48,       1;
%e A141690   1,  109,   594,    594,     109,       1;
%e A141690   1,  234,  2367,   4812,    2367,     234,      1;
%e A141690   1,  487,  8565,  31203,   31203,    8565,    487,     1;
%e A141690   1,  996, 29188, 176412,  312310,  176412,  29188,   996,    1;
%e A141690   1, 2017, 95644, 910300, 2620582, 2620582, 910300, 95644, 2017, 1;
%p A141690 A141690 := proc(n,m)
%p A141690         2*A008292(n+1,m+1)-binomial(n,m) ;
%p A141690 end proc: # _R. J. Mathar_, Jul 12 2012
%t A141690 Table[Table[(2*Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}] - Binomial[n - 1, k]), {k, 0, n - 1}], {n, 1, 10}]; Flatten[%]
%Y A141690 Cf. A008292, A007318.
%K A141690 nonn,tabl
%O A141690 0,5
%A A141690 _Roger L. Bagula_, Sep 09 2008