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%I A323833 #45 Jun 11 2025 06:46:56
%S A323833 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,
%T A323833 -24,-45,-61,-272,-333,-378,-402,-402,-378,-333,-272,-1385,-1113,-780,
%U A323833 -402,0,402,780,1113,1385,7936,9321,10434,11214,11616,11616,11214,10434,9321,7936
%N A323833 A Seidel matrix A(n,k) read by antidiagonals upwards.
%C A323833 The first row is a signed version of the Euler numbers A000111.
%C A323833 Other rows are defined by A(n+1,k) = A(n,k) + A(n,k+1).
%H A323833 Alois P. Heinz, <a href="/A323833/b323833.txt">Antidiagonals n = 0..140, flattened</a>
%H A323833 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 a_{n,k} on p. 18).
%F A323833 From _Petros Hadjicostas_, Mar 04 2021: (Start)
%F A323833 Formulas about the square array A(n,k) (n,k > 0):
%F A323833 A(n,0) = -A163747(n) = (-1)^(n+1)*A(0,n) = if(n==0, 0, (-1)^floor(n/2)*A000111(n)).
%F A323833 A(n,n) = 0 and A(n,k) + (-1)^(n+k)*A(k,n) = 0.
%F A323833 A(n, k) = Sum_{i=0..n} binomial(n, i)*A(0,k+i).
%F A323833 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).
%F A323833 Formulas about the triangular array T(n,k) = A(n-k,k) (0 <= k <= n):
%F A323833 T(n+1,k+1) = T(n+1,k) - T(n,k).
%F A323833 T(n,k) = -(-1)^n*T(n,n-k).
%F A323833 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)
%e A323833 Triangular array T(n,k) = A(n-k,k) (n >= 0, k = 0..n), read from the antidiagonals upwards of square array A:
%e A323833 0;
%e A323833 1, 1;
%e A323833 1, 0, -1;
%e A323833 -2, -3, -3, -2;
%e A323833 -5, -3, 0, 3, 5;
%e A323833 16, 21, 24, 24, 21, 16;
%e A323833 61, 45, 24, 0, -24, -45, -61;
%e A323833 -272, -333, -378, -402, -402, -378, -333, -272;
%e A323833 ...
%e A323833 From _Petros Hadjicostas_, Mar 04 2021: (Start)
%e A323833 Square array A(n,k) (n, k >= 0) begins:
%e A323833 0, 1, -1, -2, 5, 16, -61, -272, 1385, ...
%e A323833 1, 0, -3, 3, 21, -45, -333, 1113, 9321, ...
%e A323833 1, -3, 0, 24, -24, -378, 780, 10434, -33264, ...
%e A323833 -2, -3, 24, 0, -402, 402, 11214, -22830, -480162, ...
%e A323833 -5, 21, 24, -402, 0, 11616, -11616, -502992, 1017600, ...
%e A323833 16, 45, -378, -402, 11616, 0, -514608, 514608, 31880016, ...
%e A323833 ... (End)
%p A323833 A323833 := proc(n,k)
%p A323833 option remember;
%p A323833 local i ;
%p A323833 if k =0 then
%p A323833 -A163747(n) ;
%p A323833 elif n =0 then
%p A323833 (-1)^k*A163747(k) ;
%p A323833 elif k =n then
%p A323833 0 ;
%p A323833 else
%p A323833 add(binomial(n,i)*procname(0,k+i), i=0..n) ;
%p A323833 end if;
%p A323833 end proc:
%p A323833 seq(seq(A323833(d-k,k),k=0..d),d=0..12) ; # _R. J. Mathar_, Jun 11 2025
%o A323833 (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 A323833 c(n) = if(n==0, 0, (-1)^floor(n/2)*b(n))
%o A323833 A(n, k) = sum(i=0, n, binomial(n, i)*c(k+i)) \\ _Petros Hadjicostas_, Mar 04 2021
%Y A323833 Cf. A000111, A002832 (next-to-main diagonal), A163747, A323834.
%K A323833 sign,tabl
%O A323833 0,7
%A A323833 _N. J. A. Sloane_, Feb 03 2019
%E A323833 More terms from _Alois P. Heinz_, Feb 09 2019