OFFSET
0,5
COMMENTS
Compare to triangle A108558, where row n equals the (n+1)-th differences of the crystal ball sequence for D_n lattice.
LINKS
Muniru A Asiru, Rows n=0..100 of triangle, flattened
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
FORMULA
T(n, k) = C(2*n+1, 2*k) - 2*n*C(n-1, k-1).
Row sums are 2^n*(2^n - n) for n >= 0.
G.f. for coordination sequence of B_n lattice: (Sum_{i=0..n} binomial(2*n+1, 2*i)*z^i) - 2*n*z*(1+z)^(n-1))/(1-z)^n. [Bacher et al.]
EXAMPLE
G.f.s of initial rows of square array A108998 are:
(1),
(1 + x)/(1-x),
(1 + 6*x + x^2)/(1-x)^2;
(1 + 15*x + 23*x^2 + x^3)/(1-x)^3;
(1 + 28*x + 102*x^2 + 60*x^3 + x^4)/(1-x)^4.
Triangle begins:
1;
1, 1;
1, 6, 1;
1, 15, 23, 1;
1, 28, 102, 60, 1;
1, 45, 290, 402, 125, 1;
1, 66, 655, 1596, 1167, 226, 1;
1, 91, 1281, 4795, 6155, 2793, 371, 1;
1, 120, 2268, 12040, 23750, 18888, 5852, 568, 1;
1, 153, 3732, 26628, 74574, 91118, 49380, 11124, 825, 1;
MATHEMATICA
T[n_, k_] := Binomial[2n+1, 2k] - 2n * Binomial[n-1, k-1]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Dec 14 2018 *)
PROG
(PARI) T(n, k)=binomial(2*n+1, 2*k)-2*n*binomial(n-1, k-1)
(GAP) Flat(List([0..10], n->List([0..n], k->Binomial(2*n+1, 2*k)-2*n*Binomial(n-1, k-1)))); # Muniru A Asiru, Dec 14 2018
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jun 17 2005
STATUS
approved