A336859 - OEIS

%I #24 Aug 09 2020 13:39:37

%S 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,

%T 1,1,2321,1805,897,321,81,13,1,1,10879,8557,4431,1729,511,109,15,1,1,

%U 52465,41585,22149,9161,3001,761,141,17,1,1,258563,206097,112047,48313,17003,4841,1079,177,19,1

%N Mirror image of triangular array A336858.

%C This is a mirror image of A336858, which is a shifted version of _J. M. Bergot_'s triangular array first described in A104858.

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

%F 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.

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

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

%F 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).

%F 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).

%e Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:

%e   1;

%e   1,     1;

%e   1,     3,    1;

%e   1,     9,    5,    1;

%e   1,    31,   21,    7,    1;

%e   1,   121,   89,   37,    9,   1;

%e   1,   515,  393,  183,   57,  11,   1;

%e   1,  2321, 1805,  897,  321,  81,  13,  1;

%e   1, 10879, 8557, 4431, 1729, 511, 109, 15, 1;

%e   ...

%o (PARI) A000108(n) = binomial(2*n, n)/(n+1);

%o A086616(n) = sum(k=0, n, binomial(n+k+1, 2*k+1) * A000108(k));

%o 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))));

%o for(n=0, 10, for (k=0, n, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Aug 08 2020

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

%K nonn,tabl

%O 0,5

%A _Petros Hadjicostas_, Aug 05 2020