A323834 A Seidel matrix A(n,k) read by antidiagonals downwards.

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, -1385, -1657, -1868, -2002, -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

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

Table of n, a(n) for n=0..65.

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

Cf. A000111, A000657 (next-to-main diagonal), A323833.

Sequence in context: A017838 A181567 A058294 * A082868 A219539 A154556

Adjacent sequences: A323831 A323832 A323833 * A323835 A323836 A323837

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