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.

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

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

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: A176152 A146958 A154653 * A203005 A357156 A296963

Adjacent sequences: A108998 A108999 A109000 * A109002 A109003 A109004

Keyword

nonn,tabl

Author

Paul D. Hanna, Jun 17 2005

Status

approved