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 A262227 Eulerian numbers of type D, the complementary type. 1

%I

%S 0,0,1,0,4,0,0,13,10,1,0,40,112,40,0,0,121,836,846,116,1,0,364,5264,

%T 11784,5264,364,0,0,1093,30318,129879,129844,30339,1086,1,0,3280,

%U 165792,1242672,2337472,1242672,165792,3280,0,0,9841,878152,10854028,34706584,34706710,10853944,878188,9832,1

%N Eulerian numbers of type D, the complementary type.

%C Named D~(n, k) (the complementary type D triangle) in the Borowiec link.

%H Anna Borowiec, Wojciech Mlotkowski, <a href="http://arxiv.org/abs/1509.03758">New Eulerian numbers of type D</a>, arXiv:1509.03758 [math.CO], 2015.

%F T(n, k) = (A060187(n+1, k+1) - (-1)^k*binomial(n, k))/2.

%e Triangle begins:

%e 0;

%e 0, 1;

%e 0, 4, 0;

%e 0, 13, 10, 1;

%e 0, 40, 112, 40, 0;

%e 0, 121, 836, 846, 116, 1;

%e 0, 364, 5264, 11784, 5264, 364, 0;

%e ...

%o (PARI) B(n, k) = if( n<k || k<1, 0, sum(i=1, k, (-1)^(k-i) * binomial(n, k-i) * (2*i-1)^(n-1)));

%o T(n, k) = (A060187(n+1, k+1) - (-1)^k*binomial(n, k))/2;

%Y Cf. A060187, A262226.

%K nonn,tabl

%O 0,5

%A _Michel Marcus_, Sep 15 2015

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