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

Author

Paul D. Hanna, Jun 17 2005

Keywords

• nonn
• tabl

Comments

Compare to triangle A108558, where row n equals the (n+1)-th differences of the crystal ball sequence for D_n lattice.

Examples

			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;
		

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.

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).

Programs

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

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 *)

PARI

T(n,k)=binomial(2*n+1,2*k)-2*n*binomial(n-1,k-1)

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.]