cp's OEIS Frontend

A099023 Diagonal of Euler-Seidel matrix with start sequence e.g.f. 1-tanh(x).

%I A099023 #21 Jun 11 2019 07:00:14
%S A099023 1,-1,4,-46,1024,-36976,1965664,-144361456,13997185024,-1731678144256,
%T A099023 266182076161024,-49763143319190016,11118629668610842624,
%U A099023 -2925890822304510631936,895658946905031792553984
%N A099023 Diagonal of Euler-Seidel matrix with start sequence e.g.f. 1-tanh(x).
%C A099023 T(2n,n), where T is A008280 (signed).
%H A099023 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>
%F A099023 |a(n)| = A000657(n) - _Sean A. Irvine_, Dec 22 2010
%F A099023 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 A099023 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 A099023 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 A099023 A099023List[15] (* _Jean-François Alcover_, Jun 11 2019, after _Peter Luschny_ *)
%o A099023 (Sage) # Variant of an algorithm of L. Seidel (1877).
%o A099023 def A099023_list(n) :
%o A099023     dim = 2*n; E = matrix(ZZ, dim); E[0,0] = 1
%o A099023     for m in (1..dim-1) :
%o A099023         if m % 2 == 0 :
%o A099023             E[m,0] = 1;
%o A099023             for k in range(m-1,-1,-1) :
%o A099023                 E[k,m-k] = E[k+1,m-k-1] - E[k,m-k-1]
%o A099023         else :
%o A099023             E[0,m] = 1;
%o A099023             for k in range(1,m+1,1) :
%o A099023                 E[k,m-k] = E[k-1,m-k+1] + E[k-1,m-k]
%o A099023     return [E[k,k] for k in range((dim+1)//2)]
%o A099023 # _Peter Luschny_, Jul 14 2012
%Y A099023 Cf. A000657, A008280, A029582, A009744.
%K A099023 sign
%O A099023 0,3
%A A099023 _Ralf Stephan_, Sep 23 2004