A099023 - OEIS

%I #21 Jun 11 2019 07:00:14

%S 1,-1,4,-46,1024,-36976,1965664,-144361456,13997185024,-1731678144256,

%T 266182076161024,-49763143319190016,11118629668610842624,

%U -2925890822304510631936,895658946905031792553984

%N Diagonal of Euler-Seidel matrix with start sequence e.g.f. 1-tanh(x).

%C T(2n,n), where T is A008280 (signed).

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>

%F |a(n)| = A000657(n) - _Sean A. Irvine_, Dec 22 2010

%F G.f.: 1/G(0) where G(k) = 1 + x*(k+1)*(4*k+1)/(1 + x*(k+1)*(4*k+3)/G(k+1) ) ;  (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Feb 05 2013

%F G.f.: G(0)/(1+x), where G(k) = 1 - x^2*(k+1)^2*(4*k+1)*(4*k+3)/( x^2*(k+1)^2*(4*k+1)*(4*k+3) - (1 + x*(8*k^2+4*k+1))*(1 + x*(8*k^2+20*k+13))/G(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Feb 01 2014

%t A099023List[n_] := Module[{e, dim, m, k}, dim = 2 n; e[0, 0] = 1; For[m = 1, m <= dim - 1, m++, If[EvenQ[m], e[m, 0] = 1; For[k = m - 1, k >= -1, k--, e[k, m - k] = e[k + 1, m - k - 1] - e[k, m - k - 1]], e[0, m] = 1; For[k = 1, k <= m + 1, k++, e[k, m - k] = e[k - 1, m - k + 1] + e[k - 1, m - k]]]]; Table[e[k, k], {k, 0, (dim + 1)/2 - 1}]];

%t A099023List[15] (* _Jean-François Alcover_, Jun 11 2019, after _Peter Luschny_ *)

%o (Sage) # Variant of an algorithm of L. Seidel (1877).

%o def A099023_list(n) :

%o     dim = 2*n; E = matrix(ZZ, dim); E[0,0] = 1

%o     for m in (1..dim-1) :

%o         if m % 2 == 0 :

%o             E[m,0] = 1;

%o             for k in range(m-1,-1,-1) :

%o                 E[k,m-k] = E[k+1,m-k-1] - E[k,m-k-1]

%o         else :

%o             E[0,m] = 1;

%o             for k in range(1,m+1,1) :

%o                 E[k,m-k] = E[k-1,m-k+1] + E[k-1,m-k]

%o     return [E[k,k] for k in range((dim+1)//2)]

%o # _Peter Luschny_, Jul 14 2012

%Y Cf. A000657, A008280, A029582, A009744.

%K sign

%O 0,3

%A _Ralf Stephan_, Sep 23 2004