A323833 A Seidel matrix A(n,k) read by antidiagonals upwards.

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

Author

N. J. A. Sloane, Feb 03 2019

Keywords

• sign
• tabl

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).

Examples

			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)
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• 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).

Crossrefs

Cf. A000111, A002832 (next-to-main diagonal), A163747, A323834.

Programs

Maple

A323833 := proc(n,k)
    option remember;
    local i ;
    if k =0 then
        -A163747(n) ;
    elif n =0 then
        (-1)^k*A163747(k) ;
    elif k =n then
        0 ;
    else
        add(binomial(n,i)*procname(0,k+i), i=0..n) ;
    end if;
end proc:
seq(seq(A323833(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Jun 11 2025

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

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)

Extensions

More terms from Alois P. Heinz, Feb 09 2019