A336859 Mirror image of triangular array A336858.

1, 1, 1, 1, 3, 1, 1, 9, 5, 1, 1, 31, 21, 7, 1, 1, 121, 89, 37, 9, 1, 1, 515, 393, 183, 57, 11, 1, 1, 2321, 1805, 897, 321, 81, 13, 1, 1, 10879, 8557, 4431, 1729, 511, 109, 15, 1, 1, 52465, 41585, 22149, 9161, 3001, 761, 141, 17, 1, 1, 258563, 206097, 112047, 48313, 17003, 4841, 1079, 177, 19, 1

Offset: 0

Author

Petros Hadjicostas, Aug 05 2020

Keywords

• nonn
• tabl

Comments

This is a mirror image of A336858, which is a shifted version of J. M. Bergot's triangular array first described in A104858.

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,     1;
  1,     3,    1;
  1,     9,    5,    1;
  1,    31,   21,    7,    1;
  1,   121,   89,   37,    9,   1;
  1,   515,  393,  183,   57,  11,   1;
  1,  2321, 1805,  897,  321,  81,  13,  1;
  1, 10879, 8557, 4431, 1729, 511, 109, 15, 1;
  ...
		

Crossrefs

Cf. A001003, A006318, A086616, A104858, A333090, A336858.

Programs

PARI

A000108(n) = binomial(2*n, n)/(n+1);
A086616(n) = sum(k=0, n, binomial(n+k+1, 2*k+1) * A000108(k));
T(n, k) = if ((k==0) || (n==k), 1, if ((n<0) || (k<0), 0, if (k==1, A086616(n-1), if (n>k, T(n, k-1) - T(n-1, k-1) - T(n-1, k-2), 0))));
for(n=0, 10, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Aug 08 2020

Formula

T(n,k) = A336858(n, n-k) for 0 <= k <= n.

T(n,k) = T(n, k-1) - T(n-1, k-1) - T(n-1, k-2) for 2 <= k <= n with T(n,0) = T(n,n) = 1 for n >= 0 and T(n,1) = A086616(n-1) for n >= 1.

T(2*n,n) = A333090(n).

Sum_{k=0..n} T(n,k) = A104858(n) for n >= 0.

Bivariate o.g.f.: (x*y*(1 + g(x)) + 1 - y)/((1 - x)*(1 - y + x*y + x*y^2)), where g(w) = 2/(1 - w + sqrt(1 - 6*w + w^2)) = o.g.f. of A006318 (large Schroeder numbers).

Bivariate o.g.f.: (2*x*y*q(x) + 1 - y)/((1 - x)*(1 - y + x*y + x*y^2)), where q(w) = 2/(1 + w + sqrt(1 - 6*w + w^2)) = o.g.f. of A001003 (little Schroeder numbers).