A transform of

[Gottfried Helms]

Dear list -

 I am currently analyzing the matrix G =

  1  -1  0   0  0   0   0    0   0    0
  .  -1  2  -1  0   0   0    0   0    0
  .   .  1  -3  3  -1   0    0   0    0
  .   .  .  -1  4  -6   4   -1   0    0
  .   .  .   .  1  -5  10  -10   5   -1
  .   .  .   .  .  -1   6  -15  20  -15
  .   .  .   .  .   .   1   -7  21  -35
  .   .  .   .  .   .   .   -1   8  -28
  .   .  .   .  .   .   .    .   1   -9
  .   .  .   .  .   .   .    .   .   -1

which plays seemingly a multitalent role as generator
for basic number-theoretical sequences A() as

  - bernoulli-numbers
  - "eta"-numbers
  - and all sequences like 1/binomial(k,n) (with varying k and fixed n)

if the sequence A() in question is taken as column-vector
scaled with its (column-index+1). OEIS: [1]

For instance , taking the column-vector of scaled
bernoulli-numbers

 B = [  b0*1,   b1*2,  b2*3,... ] (where b1 = +1/2)
   = [   1*1,  1/2*2, 1/6*3,... ]

then

  G * B = 0

as well as if I use

 B = [  a0*1,  a1*2,  a2*3,        ... ] (where a_k= "eta"-numbers)
   = [   1*1, 1/2*2,   0*3, -1/4*4, ...]

  G * B = 0

as well as any of the reciprocals of the binomials

 B = [  a0*1,  a1*2,  a2*3,        ... ] (where a_k= binomial(k+1,1))
   = [   1*1, 1/2*2, 1/3*3,  1/4*4, ...]

 B = [  a0*1,  a1*2,   a2*3,        ... ] (where a_k= binomial(k+3,2))
   = [ 1/3*1, 1/6*2, 1/10*3, 1/15*4, ...]
                                            and so on
with the limit of

 B = [  a0*1,  a1*2,   a2*3,        ... ] (where a_k= 1/2^k)
   = [   1*1, 1/2*2,  1/4*3,  1/8*4, ...]


all

  G * B = 0

means: all these very different sequences are eigenvectors of G.
(and whose generation matrix G allows to recursively compute
Bernoulli-, eta and the binomial-sequences by the same recursion
formula only using different parameters)

----------------------

I think, this alone is a remarkable property.

An additional surprise is to look at the inverse; this is

H = G^-1 =

  1  -1  2  -5  14  -42  132  -429  1430  -4862
  .  -1  2  -5  14  -42  132  -429  1430  -4862
  .   .  1  -3   9  -28   90  -297  1001  -3432
  .   .  .  -1   4  -14   48  -165   572  -2002
  .   .  .   .   1   -5   20   -75   275  -1001
  .   .  .   .   .   -1    6   -27   110   -429
  .   .  .   .   .    .    1    -7    35   -154
  .   .  .   .   .    .    .    -1     8    -44
  .   .  .   .   .    .    .     .     1     -9
  .   .  .   .   .    .    .     .     .     -1

which is also in OEIS as the (transposed) triangle of
Catalan-numbers [2].

I tried to see, what the rowsums are; these are divergent,
but not too bad, so they can be Euler-summed. I approximated
it to

   [r,r^2,r^3,..]   where r=0.61..., the golden ratio

Having this amazing result I tried to find a general
expression for the transformation of the rows of this
triangle on a powerseries and I seem to have arrived at
a true powerseries-to-powerseries-transformation.
Assume a powerseries-vector V(x) = [1,x,x^2,x^3,...]~
then

  H *   x * V(x) =  y * V(y)

  where
      y =sqrt( x + 1/4 ) - 1/2


for instance, if x = 3/4 then y = 1/2 and

  H * [3/4, 9/16, 27/64... ] = [ 1/2 , 1/4, 1/8, ....]

Don't know, whether this is worth to be mentioned
in the database...

Gottfried Helms


-----------------------------


[1] A030528 Triangle read by rows: a(n,k)=binom(k,n-k).
    The signed triangular matrix a(n,m)*(-1)^(n-m) is
    the inverse matrix of the triangular Catalan convolution
    matrix A033184(n+1,m+1),
    n >= m >= 0, with A033184(n,m) := 0 if n<m.
     Row sums A000045(n+1) (Fibonacci).

[2] A033184:  Catalan triangle A009766 transposed.
    http://www.research.att.com/~njas/sequences/A033184


[3]
Concerning the columnsum-/alternating columnsum-property of G and the
Fibonacci-relation:
 - see also A030528, crossrefs 1

 - [A010892]	http://www.research.att.com/~njas/sequences/A010892
   A010892:Inverse of 6th cyclotomic polynomial. A period 6 sequence.
      :1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1
   COMMENT :Any sequence b(n) satisfying the recurrence b(n)=b(n-1)-b(n-2) can be written as
          b(n) = b(0) * a(n)  +  (b(1)-b(0)) * a(n-1)  . (...)
   REFERENCES :Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences,
   Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.