A122897 Riordan array (1/(1-x), c(x)-1) where c(x) is the g.f. of A000108.

1, 1, 1, 1, 3, 1, 1, 8, 5, 1, 1, 22, 19, 7, 1, 1, 64, 67, 34, 9, 1, 1, 196, 232, 144, 53, 11, 1, 1, 625, 804, 573, 261, 76, 13, 1, 1, 2055, 2806, 2211, 1171, 426, 103, 15, 1, 1, 6917, 9878, 8399

Offset

0,5

Comments

Product of A007318 and A122896. Inverse of Riordan array ((1+x+x^2)/(1+x)^2,x/(1+x)^2). Row sums are A024718.

The n-th row polynomial (in descending powers of x) equals the n-th Taylor polynomial of the rational function (1 - x^2)/(1 + x + x^2) * (1 + x)^(2*n) about 0. For example, for n = 4 we have (1 - x^2)/( 1 + x + x^2) * (1 + x)^8 = (x^4 + 22*x^3 + 19*x^2 + 7*x + 1) + O(x^5). - Peter Bala, Feb 21 2018

Links

Table of n, a(n) for n=0..48.

P. Bala, A 4-parameter family of embedded Riordan arrays

Formula

T(n,k) = binomial(2*n,n-k) + 2*Sum_{j = 1..n-k} cos((2/3)*Pi*j)* binomial(2*n, n-k-j). - Peter Bala, Feb 21 2018

T(n,k) = k*Sum_{i=0..n-k} C(2*(i+k),i)/(i+k), T(n,0)=1. - Vladimir Kruchinin, Jun 13 2020

Example

Triangle begins
  1,
  1,     1,
  1,     3,     1,
  1,     8,     5,     1,
  1,    22,    19,     7,     1,
  1,    64,    67,    34,     9,    1,
  1,   196,   232,   144,    53,   11,    1,
  1,   625,   804,   573,   261,   76,   13,   1,
  1,  2055,  2806,  2211,  1171,  426,  103,  15,   1,
  1,  6917,  9878,  8399,  4979, 2126,  647, 134,  17,  1,
  1, 23713, 35072, 31655, 20483, 9878, 3554, 932, 169, 19, 1

Maple

A122897 := proc (n, k)
  binomial(2*n, n-k) + 2*add(cos((2/3)*Pi*j)*binomial(2*n, n-k-j), j = 1..n-k)
end proc:
for n from 0 to 10 do
seq(A122897(n, k), k = 0..n)
end do; # Peter Bala, Feb 21 2018

Crossrefs

Sequence in context: A114276 A152879 A098747 * A117425 A287215 A168216

Adjacent sequences: A122894 A122895 A122896 * A122898 A122899 A122900

Keyword

easy,nonn,tabl

Author

Paul Barry, Sep 18 2006

Status

approved