OFFSET
0,5
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 b_{n,k} on p. 19).
FORMULA
From Petros Hadjicostas, Mar 02 2021: (Start)
Formulas for the square array A(n,k) (n, k >= 0):
A(0,k) = (-1)^floor((k-1)/2)*A000111(k) for k > 0 with A(0,0) = 0.
A(n,k) = Sum_{i=0..n} binomial(n, i)*A(0,k+i) for n, k >= 0.
A(n,n)/2 = A(n+1,n) = +/- A000657(n) for n > 0.
Bivariate e.g.f.: Sum_{n,k >= 0} A(n,k)*(x^n/n!)*(y^k/k!) = (-sech(x + y) + tanh(x + y) + 1)*exp(x).
Formulas for the triangular array T(n,k) = A(k,n-k) (n >= 0, 0 <= k <= n):
T(n,k) = T(n-1,k-1) + T(n,k-1) for 1 <= k <= n with T(n,0) = (-1)^floor((n-1)/2) * A000111(n) for n > 0 and T(0,0) = 0.
T(n,k) = Sum_{i=0..k} binomial(k,i)*T(n-k+i,0) for 0 <= k <= n. (End)
EXAMPLE
Read as triangle T(n,k) = A(k, n-k) (n >= 0, k = 0..n), the first few antidiagonals of the square array A are:
0,
1, 1,
1, 2, 3,
-2, -1, 1, 4,
-5, -7, -8, -7, -3,
16, 11, 4, -4, -11, -14,
61, 77, 88, 92, 88, 77, 63,
-272, -211, -134, -46, 46, 134, 211, 274,
...
From Petros Hadjicostas, Mar 02 2021: (Start)
Square array A(n,k) (with rows n >= 0 and columns k >= 0) begins:
0, 1, 1, -2, -5, 16, 61, -272, -1385, ...
1, 2, -1, -7, 11, 77, -211, -1657, 6551, ...
3, 1, -8, 4, 88, -134, -1868, 4894, 65008, ...
4, -7, -4, 92, -46, -2002, 3026, 69902, -179806, ...
-3, -11, 88, 46, -2048, 1024, 72928, -109904, -3784448, ...
-14, 77, 134, -2002, -1024, 73952, -36976, -3894352, 5860016, ...
... (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-1)/2)*b(n))
A(n, k) = sum(i=0, n, binomial(n, i)*c(k+i)) \\ Petros Hadjicostas, Mar 02 2021
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
N. J. A. Sloane, Feb 03 2019
EXTENSIONS
Typo corrected by and more terms from Petros Hadjicostas, Mar 02 2021
STATUS
approved