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

Author

Paul Barry, Sep 18 2006

Keywords

• easy
• nonn
• tabl

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

Examples

			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
		

Links

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

Programs

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

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