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A263985 Triangle of signed Eulerian numbers on involutions, read by rows.

%I A263985 #9 Sep 26 2018 16:26:04
%S A263985 1,-1,1,-1,-2,1,1,-2,-2,1,1,6,0,-2,1,-1,3,14,2,-3,1,-1,-12,-15,12,-1,
%T A263985 -4,1,1,-4,-51,-76,4,-3,-4,1,1,20,67,-10,-80,30,3,-4,1,-1,5,137,517,
%U A263985 414,66,75,7,-5,1,-1,-30,-192,-140,721,588,-49,44,0,-6,1
%N A263985 Triangle of signed Eulerian numbers on involutions, read by rows.
%H A263985 M. Barnabei, F. Bonetti, M. Silimbani, <a href="http://puma.dimai.unifi.it/19_2_3/11.pdf">The signed Eulerian numbers on involutions</a>, PU. M. A. Vol. 19 (2008) pp. 117-126.
%H A263985 M. Barnabei, F. Bonetti, M. Silimbani, <a href="http://arxiv.org/abs/0803.2126">The signed Eulerian numbers on involutions</a>, arXiv:0803.2126 [math.CO], 2008.
%H A263985 J. Desarmenien and D. Foata, <a href="http://dx.doi.org/10.1016/0012-365X(92)90364-L">The signed Eulerian numbers</a>, Discrete Math. 99 (1992), no. 1-3, 49-58.
%F A263985 T(n, k) = Sum_{m=0..k+1} (-1)^(k-m+1)*C(n+1,k-m+1)*Sum_{j=0..floor(n/2)} (-1)^j*C(C(m+1,2)+j-1,j)*C(m,n-2*j);
%e A263985 Triangle begins:
%e A263985 1;
%e A263985 -1, 1;
%e A263985 -1, -2, 1;
%e A263985 1, -2, -2, 1;
%e A263985 1, 6, 0, -2, 1;
%e A263985 -1, 3, 14, 2, -3, 1;
%e A263985 -1, -12, -15, 12, -1, -4, 1;
%e A263985 ...
%t A263985 T[n_, k_] := Sum[(-1)^(k-m+1) Binomial[n+1, k-m+1] Sum[(-1)^j Binomial[ Binomial[m+1, 2]+j-1, j] Binomial[m, n-2j], {j, 0, n/2}], {m, 0, k+1}];
%t A263985 Table[T[n, k], {n, 1, 11}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Sep 26 2018 *)
%o A263985 (PARI) T(n, k) = sum(m=0, k+1, (-1)^(k-m+1)*binomial(n+1,k-m+1)*sum(j=0,n\2, (-1)^j*binomial(binomial(m+1,2)+j-1,j)*binomial(m,n-2*j)));
%Y A263985 Cf. A049061.
%K A263985 sign,tabl
%O A263985 1,5
%A A263985 _Michel Marcus_, Oct 31 2015