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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 (list; table; graph; refs; listen; history; text; internal format)
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

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).

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 A131307

Adjacent sequences:  A323830 A323831 A323832 * A323834 A323835 A323836

KEYWORD

sign,tabl

AUTHOR

N. J. A. Sloane, Feb 03 2019

EXTENSIONS

More terms from Alois P. Heinz, Feb 09 2019

STATUS

approved

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Last modified July 28 19:33 EDT 2021. Contains 346335 sequences. (Running on oeis4.)