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A323833
A Seidel matrix A(n,k) read by antidiagonals upwards.
5
0, 1, 1, 1, 0, -1, -2, -3, -3, -2, -5, -3, 0, 3, 5, 16, 21, 24, 24, 21, 16, 61, 45, 24, 0, -24, -45, -61, -272, -333, -378, -402, -402, -378, -333, -272, -1385, -1113, -780, -402, 0, 402, 780, 1113, 1385, 7936, 9321, 10434, 11214, 11616, 11616, 11214, 10434, 9321, 7936
OFFSET
0,7
COMMENTS
The first row is a signed version of the Euler numbers A000111.
Other rows are defined by A(n+1,k) = A(n,k) + A(n,k+1).
LINKS
A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites classiques de nombres, Adv. Appl. Math. 17 (1996), 1-26. See Section 6 (matrix a_{n,k} on p. 18).
FORMULA
From Petros Hadjicostas, Mar 04 2021: (Start)
Formulas about the square array A(n,k) (n,k > 0):
A(n,0) = -A163747(n) = (-1)^(n+1)*A(0,n) = if(n==0, 0, (-1)^floor(n/2)*A000111(n)).
A(n,n) = 0 and A(n,k) + (-1)^(n+k)*A(k,n) = 0.
A(n, k) = Sum_{i=0..n} binomial(n, i)*A(0,k+i).
Joint e.g.f.: Sum_{n,k >= 0} A(n,k)*(x^n/n!)*(y^k/k!) = 2*exp(-y)*(1 - exp(-x - y)) / (1 + exp(-2*(x + y))) = 2*exp(x)*(exp(x+y) - 1) / (exp(2*(x+y)) + 1).
Formulas about the triangular array T(n,k) = A(n-k,k) (0 <= k <= n):
T(n+1,k+1) = T(n+1,k) - T(n,k).
T(n,k) = -(-1)^n*T(n,n-k).
T(n,k) = Sum_{i=0..n-k} binomial(n-k,i)*T(k+i,k+i) for k=0..n with initial condition T(n,n) = (-1)^n*A163747(n). (End)
EXAMPLE
Triangular array T(n,k) = A(n-k,k) (n >= 0, k = 0..n), read from the antidiagonals upwards of square array A:
0;
1, 1;
1, 0, -1;
-2, -3, -3, -2;
-5, -3, 0, 3, 5;
16, 21, 24, 24, 21, 16;
61, 45, 24, 0, -24, -45, -61;
-272, -333, -378, -402, -402, -378, -333, -272;
...
From Petros Hadjicostas, Mar 04 2021: (Start)
Square array A(n,k) (n, k >= 0) begins:
0, 1, -1, -2, 5, 16, -61, -272, 1385, ...
1, 0, -3, 3, 21, -45, -333, 1113, 9321, ...
1, -3, 0, 24, -24, -378, 780, 10434, -33264, ...
-2, -3, 24, 0, -402, 402, 11214, -22830, -480162, ...
-5, 21, 24, -402, 0, 11616, -11616, -502992, 1017600, ...
16, 45, -378, -402, 11616, 0, -514608, 514608, 31880016, ...
... (End)
PROG
(PARI) {b(n) = local(v=[1], t); if( n<0, 0, for(k=2, n+2, t=0; v = vector(k, i, if( i>1, t+= v[k+1-i]))); v[2])}; \\ Michael Somos's PARI program for A000111.
c(n) = if(n==0, 0, (-1)^floor(n/2)*b(n))
A(n, k) = sum(i=0, n, binomial(n, i)*c(k+i)) \\ Petros Hadjicostas, Mar 04 2021
CROSSREFS
Cf. A000111, A002832 (next-to-main diagonal), A163747, A323834.
Sequence in context: A008985 A326699 A138652 * A131899 A095174 A376676
KEYWORD
sign,tabl
AUTHOR
N. J. A. Sloane, Feb 03 2019
EXTENSIONS
More terms from Alois P. Heinz, Feb 09 2019
STATUS
approved