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

1, -1, 4, -46, 1024, -36976, 1965664, -144361456, 13997185024, -1731678144256, 266182076161024, -49763143319190016, 11118629668610842624, -2925890822304510631936, 895658946905031792553984

Offset: 0

Author

Ralf Stephan, Sep 23 2004

Keywords

• sign

Comments

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

Links

• Peter Luschny, An old operation on sequences: the Seidel transform

Crossrefs

Cf. A000657, A008280, A029582, A009744.

Programs

Mathematica

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}]];
A099023List[15] (* Jean-François Alcover, Jun 11 2019, after Peter Luschny *)

Sage

# Variant of an algorithm of L. Seidel (1877).
def A099023_list(n) :
    dim = 2*n; E = matrix(ZZ, dim); E[0,0] = 1
    for m in (1..dim-1) :
        if m % 2 == 0 :
            E[m,0] = 1;
            for k in range(m-1,-1,-1) :
                E[k,m-k] = E[k+1,m-k-1] - E[k,m-k-1]
        else :
            E[0,m] = 1;
            for k in range(1,m+1,1) :
                E[k,m-k] = E[k-1,m-k+1] + E[k-1,m-k]
    return [E[k,k] for k in range((dim+1)//2)]
# Peter Luschny, Jul 14 2012

Formula

|a(n)| = A000657(n) - Sean A. Irvine, Dec 22 2010

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

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