A214407 Triangle read by rows. The coefficients in ascending order of x^i of the polynomials p{0}(x) = 1 and p{n}(x) = Sum_{k=0..n-1; k even} binomial(n, k) * p{k}(0)*(1 - n%2 + x^(n - k)).

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 7, 0, 6, 0, 1, 0, 35, 0, 10, 0, 1, 121, 0, 105, 0, 15, 0, 1, 0, 847, 0, 245, 0, 21, 0, 1, 3907, 0, 3388, 0, 490, 0, 28, 0, 1, 0, 35163, 0, 10164, 0, 882, 0, 36, 0, 1, 202741, 0, 175815, 0, 25410, 0, 1470, 0, 45, 0, 1, 0

Offset: 0

Author

Peter Luschny, Jul 16 2012

Keywords

• nonn
• tabl

Comments

Matrix inverse of a signed variant of A119467.

Examples

			1
0, 1
1, 0, 1
0, 3, 0, 1
7, 0, 6, 0, 1
0, 35, 0, 10, 0, 1
121, 0, 105, 0, 15, 0, 1
0, 847, 0, 245, 0, 21, 0, 1
3907, 0, 3388, 0, 490, 0, 28, 0, 1
		

Crossrefs

Cf. A119467, A327034 (row sums), A094088 (column 0).

Programs

Sage

@CachedFunction
def A214407_poly(n, x) :
    return 1 if n==0 else add(A214407_poly(k, 0)*binomial(n, k)*(x^(n-k)+1-n%2) for k in range(n)[::2])
def A214407_row(n) :
    R = PolynomialRing(ZZ, 'x')
    return R(A214407_poly(n,x)).coeffs()
for n in (0..8) : A214407_row(n)

Formula

T(n, k) = n! * [y^k] [x^n] (exp(x * y) / (2 - cosh(x))). - Peter Luschny, May 06 2023