%I #60 Feb 22 2021 02:55:11

%S 1,0,-1,-1,-1,0,0,1,2,2,5,5,4,2,0,0,-5,-10,-14,-16,-16,-61,-61,-56,

%T -46,-32,-16,0,0,61,122,178,224,256,272,272,1385,1385,1324,1202,1024,

%U 800,544,272,0,0,-1385,-2770,-4094,-5296,-6320,-7120,-7664,-7936,-7936,-50521,-50521,-49136,-46366,-42272,-36976,-30656,-23536,-15872,-7936,0

%N The infinite Seidel matrix H read by antidiagonals upwards; H = (H(n,k): n,k >= 0).

%C This is, in essence, a signed version of the triangle in A008280.

%H D. Dumont and G. Viennot, <a href="/A110501/a110501.pdf">A combinatorial interpretation of the Seidel generation of Genocchi numbers</a>, Preprint, Annotated scanned copy.

%H D. Dumont and G. Viennot, <a href="https://doi.org/10.1016/S0167-5060(08)70696-4">A combinatorial interpretation of the Seidel generation of Genocchi numbers</a>, Annals of Discrete Mathematics, 6 (1980), 77-87.

%H Dominique Foata and Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~foata/paper/pub123Seidel.pdf">Seidel Triangle Sequences and Bi-Entringer Numbers</a>, November 20, 2013.

%H Dominique Foata and Guo-Niu Han, <a href="https://doi.org/10.1016/j.ejc.2014.06.007">Seidel Triangle Sequences and Bi-Entringer Numbers</a>, European Journal of Combinatorics, 42 (2014), 243-260.

%H L. Seidel, <a href="http://publikationen.badw.de/de/003384831/pdf/CC%20BY">Über eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen</a>, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, Vol. 7 (1877), pp. 157-187; see Beilage 4 (p. 187).

%F From _Petros Hadjicostas_, Feb 20 2021: (Start)

%F H(n,0) = A122045(n).

%F H(0,k) = (-1)^n*A155585(n).

%F H(n,k) = Sum_{i=0..n} binomial(n,i)*H(0,k+i).

%F H(n,k) = Sum_{i=0..k} (-1)^i*binomial(k,i)*H(n+k-i,0).

%F H(n,n) = A099023(n).

%F Bivariate e.g.f.: Sum_{n,k>=0} H(n,k)*(x^n/n!)*(y^k/k!) = 2*exp(x)/(1 + exp(2*(x+y))).

%F H(n,k) = (-1)^(n+k)*A239005(n+k,k), where the latter is a triangle.

%F H(n,k) = -A008280(n+k,k) if ((n+k) mod 4) == 1 or 2, and H(n,k) = A008280(n+k,k) if ((n+k) mod 4) == 3 or 0, provided A008280 is read as a triangle. (End)

%e Array begins:

%e 1 -1 0 2 0 -16 0 272 0 ...

%e 0 -1 2 2 -16 -16 272 272 ...

%e -1 1 4 -14 -32 256 544 ...

%e 0 5 -10 -46 224 800 ...

%e 5 -5 -56 178 1024 ...

%e 0 -61 122 1202 ...

%e -61 61 1324 ...

%e 0 1385 ...

%e 1385 ...

%e ...

%o (PARI) a(n) = 2^n*2^(n+1)*(subst(bernpol(n+1, x), x, 3/4) - subst(bernpol(n+1, x), x, 1/4))/(n+1) /* A122045 */

%o H(n,k) = sum(i=0, k, (-1)^i*binomial(k,i)*a(n+k-i)) /* _Petros Hadjicostas_, Feb 21 2021 */

%o /* Second PARI program (same a(n) for A122045 as above) */

%o H(n,k) = (-1)^(n+k)*sum(i=0, k, binomial(k,i)*a(n+i)) /* _Petros Hadjicostas_, Feb 21 2021 */

%Y Cf. A008280, A099023, A122045, A155585, A239005.

%K sign,tabl

%O 0,9

%A _N. J. A. Sloane_, Feb 17 2014

%E More terms from _Petros Hadjicostas_, Feb 21 2021