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

0,5

Comments

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

Links

Table of n, a(n) for n=0..65.

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).

Example

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;
  ...

Prog

(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

Crossrefs

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

Sequence in context: A074881 A142992 A145905 * A171435 A144183 A050153

Adjacent sequences: A336856 A336857 A336858 * A336860 A336861 A336862

Keyword

nonn,tabl

Author

Petros Hadjicostas, Aug 05 2020

Status

approved