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%I #40 Mar 04 2021 02:37:43
%S 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,
%T 88,77,63,-272,-211,-134,-46,46,134,211,274,-1385,-1657,-1868,-2002,
%U -2048,-2002,-1868,-1657,-1383,7936,6551,4894,3026,1024,-1024,-3026,-4894,-6551,-7934,50521,58457,65008,69902,72928,73952,72928,69902,65008,58457,50523
%N A Seidel matrix A(n,k) read by antidiagonals downwards.
%C The first row is a signed version of the Euler numbers A000111.
%C Other rows are defined by A(n+1,k) = A(n,k) + A(n,k+1).
%H A. Randrianarivony and J. Zeng, <a href="http://dx.doi.org/10.1006/aama.1996.0001">Une famille de polynomes qui interpole plusieurs suites classiques de nombres</a>, Adv. Appl. Math. 17 (1996), 1-26. See Section 6 (matrix b_{n,k} on p. 19).
%F From _Petros Hadjicostas_, Mar 02 2021: (Start)
%F Formulas for the square array A(n,k) (n, k >= 0):
%F A(0,k) = (-1)^floor((k-1)/2)*A000111(k) for k > 0 with A(0,0) = 0.
%F A(n,k) = Sum_{i=0..n} binomial(n, i)*A(0,k+i) for n, k >= 0.
%F A(n,n)/2 = A(n+1,n) = +/- A000657(n) for n > 0.
%F 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).
%F Formulas for the triangular array T(n,k) = A(k,n-k) (n >= 0, 0 <= k <= n):
%F 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.
%F T(n,k) = Sum_{i=0..k} binomial(k,i)*T(n-k+i,0) for 0 <= k <= n. (End)
%e 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:
%e 0,
%e 1, 1,
%e 1, 2, 3,
%e -2, -1, 1, 4,
%e -5, -7, -8, -7, -3,
%e 16, 11, 4, -4, -11, -14,
%e 61, 77, 88, 92, 88, 77, 63,
%e -272, -211, -134, -46, 46, 134, 211, 274,
%e ...
%e From _Petros Hadjicostas_, Mar 02 2021: (Start)
%e Square array A(n,k) (with rows n >= 0 and columns k >= 0) begins:
%e 0, 1, 1, -2, -5, 16, 61, -272, -1385, ...
%e 1, 2, -1, -7, 11, 77, -211, -1657, 6551, ...
%e 3, 1, -8, 4, 88, -134, -1868, 4894, 65008, ...
%e 4, -7, -4, 92, -46, -2002, 3026, 69902, -179806, ...
%e -3, -11, 88, 46, -2048, 1024, 72928, -109904, -3784448, ...
%e -14, 77, 134, -2002, -1024, 73952, -36976, -3894352, 5860016, ...
%e ... (End)
%o (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.
%o c(n) = if(n==0, 0, (-1)^floor((n-1)/2)*b(n))
%o A(n, k) = sum(i=0, n, binomial(n, i)*c(k+i)) \\ _Petros Hadjicostas_, Mar 02 2021
%Y Cf. A000111, A000657 (next-to-main diagonal), A323833.
%K sign,tabl
%O 0,5
%A _N. J. A. Sloane_, Feb 03 2019
%E Typo corrected by and more terms from _Petros Hadjicostas_, Mar 02 2021
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