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
Examples
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)
Links
- 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).
Programs
-
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
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)
Extensions
Typo corrected by and more terms from Petros Hadjicostas, Mar 02 2021
Comments