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A109001
Triangle, read by rows, where g.f. of row n equals the product of (1-x)^n and the g.f. of the coordination sequence for root lattice B_n, for n >= 0.
4
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, 1, 190, 5805, 53544, 201810, 350196, 291410, 114600, 19629, 1150, 1
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
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
Cf. A108998, A108999, A109000, A022144 (row 2), A022145 (row 3), A022146 (row 4), A022147 (row 5), A022148 (row 6), A022149 (row 7), A022150 (row 8), A022151 (row 9), A022152 (row 10), A022153 (row 11), A022154 (row 12).
Sequence in context: A146958 A154653 A376730 * A203005 A357156 A296963
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jun 17 2005
STATUS
approved