A214407 - OEIS

%I #12 May 06 2023 09:01:45

%S 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,

%T 0,245,0,21,0,1,3907,0,3388,0,490,0,28,0,1,0,35163,0,10164,0,882,0,36,

%U 0,1,202741,0,175815,0,25410,0,1470,0,45,0,1,0

%N 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)).

%C Matrix inverse of a signed variant of A119467.

%F T(n, k) = n! * [y^k] [x^n] (exp(x * y) / (2 - cosh(x))). - _Peter Luschny_, May 06 2023

%e 1

%e 0, 1

%e 1, 0, 1

%e 0, 3, 0, 1

%e 7, 0, 6, 0, 1

%e 0, 35, 0, 10, 0, 1

%e 121, 0, 105, 0, 15, 0, 1

%e 0, 847, 0, 245, 0, 21, 0, 1

%e 3907, 0, 3388, 0, 490, 0, 28, 0, 1

%o (Sage)

%o @CachedFunction

%o def A214407_poly(n, x) :

%o     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])

%o def A214407_row(n) :

%o     R = PolynomialRing(ZZ, 'x')

%o     return R(A214407_poly(n,x)).coeffs()

%o for n in (0..8) : A214407_row(n)

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

%K nonn,tabl

%O 0,8

%A _Peter Luschny_, Jul 16 2012