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A099023 Diagonal of Euler-Seidel matrix with start sequence e.g.f. 1-tanh(x). 6
1, -1, 4, -46, 1024, -36976, 1965664, -144361456, 13997185024, -1731678144256, 266182076161024, -49763143319190016, 11118629668610842624, -2925890822304510631936, 895658946905031792553984 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

LINKS

Table of n, a(n) for n=0..14.

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

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

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-Fran├žois Alcover, Jun 11 2019, after Peter Luschny *)

PROG

(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

CROSSREFS

Cf. A000657, A008280, A029582, A009744.

Sequence in context: A234527 A126739 A191870 * A000657 A001623 A188634

Adjacent sequences:  A099020 A099021 A099022 * A099024 A099025 A099026

KEYWORD

sign,changed

AUTHOR

Ralf Stephan, Sep 23 2004

STATUS

approved

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Last modified June 19 16:48 EDT 2019. Contains 324222 sequences. (Running on oeis4.)