A262226 - OEIS

%I #24 Jul 21 2020 22:27:04

%S 1,1,0,1,2,1,1,10,13,0,1,36,118,36,1,1,116,846,836,121,0,1,358,5279,

%T 11764,5279,358,1,1,1086,30339,129844,129879,30318,1093,0,1,3272,

%U 165820,1242616,2337542,1242616,165820,3272,1,1,9832,878188,10853944,34706710,34706584,10854028,878152,9841,0

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

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

%H Michael De Vlieger, <a href="/A262226/b262226.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened)

%H Eli Bagno, David Garber, Mordechai Novick, <a href="https://arxiv.org/abs/2004.03681">The Worpitzky identity for the groups of signed and even-signed permutations</a>, arXiv:2004.03681 [math.CO], 2020.

%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.

%H Katarzyna Kril, Wojciech Mlotkowski, <a href="https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i1p27">Permutations of Type B with Fixed Number of Descents and Minus Signs</a>, Volume 26(1) of The Electronic Journal of Combinatorics, 2019.

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

%e Triangle begins:

%e   1;

%e   1,   0;

%e   1,   2,    1;

%e   1,  10,   13,     0;

%e   1,  36,  118,    36,    1;

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

%e   1, 358, 5279, 11764, 5279, 358, 1;

%e   ...

%t T[n_, k_] := (Sum[(-1)^(k-i+1)(2i-1)^n Binomial[n+1, k-i+1], {i, 1, k+1}] + (-1)^k Binomial[n, k])/2;

%t Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 24 2018 *)

%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, A262227.

%K nonn,tabl

%O 0,5

%A _Michel Marcus_, Sep 15 2015