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A236935
The infinite Seidel matrix H read by antidiagonals upwards; H = (H(n,k): n,k >= 0).
2
1, 0, -1, -1, -1, 0, 0, 1, 2, 2, 5, 5, 4, 2, 0, 0, -5, -10, -14, -16, -16, -61, -61, -56, -46, -32, -16, 0, 0, 61, 122, 178, 224, 256, 272, 272, 1385, 1385, 1324, 1202, 1024, 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
OFFSET
0,9
COMMENTS
This is, in essence, a signed version of the triangle in A008280.
LINKS
D. Dumont and G. Viennot, A combinatorial interpretation of the Seidel generation of Genocchi numbers, Preprint, Annotated scanned copy.
D. Dumont and G. Viennot, A combinatorial interpretation of the Seidel generation of Genocchi numbers, Annals of Discrete Mathematics, 6 (1980), 77-87.
Dominique Foata and Guo-Niu Han, Seidel Triangle Sequences and Bi-Entringer Numbers, November 20, 2013.
Dominique Foata and Guo-Niu Han, Seidel Triangle Sequences and Bi-Entringer Numbers, European Journal of Combinatorics, 42 (2014), 243-260.
L. Seidel, Über eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, Vol. 7 (1877), pp. 157-187; see Beilage 4 (p. 187).
FORMULA
From Petros Hadjicostas, Feb 20 2021: (Start)
H(n,0) = A122045(n).
H(0,k) = (-1)^n*A155585(n).
H(n,k) = Sum_{i=0..n} binomial(n,i)*H(0,k+i).
H(n,k) = Sum_{i=0..k} (-1)^i*binomial(k,i)*H(n+k-i,0).
H(n,n) = A099023(n).
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))).
H(n,k) = (-1)^(n+k)*A239005(n+k,k), where the latter is a triangle.
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)
EXAMPLE
Array begins:
1 -1 0 2 0 -16 0 272 0 ...
0 -1 2 2 -16 -16 272 272 ...
-1 1 4 -14 -32 256 544 ...
0 5 -10 -46 224 800 ...
5 -5 -56 178 1024 ...
0 -61 122 1202 ...
-61 61 1324 ...
0 1385 ...
1385 ...
...
MATHEMATICA
T[0, 0] := 1; T[n_, k_] /; 0<=k<=n := T[n, k] = T[n, k+(-1)^n]+(-1)^n T[n-1, k-Mod[n, 2]]; T[n_, k_] := 0; Flatten@Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Oliver Seipel, Jan 20 2026 *)
PROG
(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 */
H(n, k) = sum(i=0, k, (-1)^i*binomial(k, i)*a(n+k-i)) /* Petros Hadjicostas, Feb 21 2021 */
/* Second PARI program (same a(n) for A122045 as above) */
H(n, k) = (-1)^(n+k)*sum(i=0, k, binomial(k, i)*a(n+i)) /* Petros Hadjicostas, Feb 21 2021 */
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
N. J. A. Sloane, Feb 17 2014
EXTENSIONS
More terms from Petros Hadjicostas, Feb 21 2021
STATUS
approved