A059678 Triangle T(n,k) giving number of fixed 2 X k polyominoes with n cells (n >= 2, 1<=k<=n-1).

1, 0, 4, 0, 1, 8, 0, 0, 6, 12, 0, 0, 1, 18, 16, 0, 0, 0, 8, 38, 20, 0, 0, 0, 1, 32, 66, 24, 0, 0, 0, 0, 10, 88, 102, 28, 0, 0, 0, 0, 1, 50, 192, 146, 32, 0, 0, 0, 0, 0, 12, 170, 360, 198, 36, 0, 0, 0, 0, 0, 1, 72, 450, 608, 258, 40, 0, 0, 0, 0, 0, 0, 14, 292, 1002, 952, 326, 44, 0, 0, 0

Offset: 2

Author

N. J. A. Sloane, Feb 05 2001

Keywords

• nonn
• easy
• nice
• tabl

Examples

			Triangle begins:
1;
0, 4;
0, 1, 8;
0, 0, 6, 12;
0, 0, 1, 18, 16;
0, 0, 0,  8, 38, 20;
0, 0, 0,  1, 32, 66, 24;
...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 2..1276
• R. C. Read, Contributions to the cell growth problem, Canad. J. Math., 14 (1962), 1-20.

Crossrefs

Column sums are A034182.

Cf. A059679, A059680, A059681, A059682.

Programs

Maple

with(combinat): for n from 2 to 30 do for k from 1 to n-1 do printf(`%d,`,sum(binomial(n-k+1, 2*k-n-v)*binomial(n-k+v, n-k), v=0..k) ) od:od:

Mathematica

t[n_, k_] := Sum[Binomial[n-k+1, 2*k-n-v]*Binomial[n-k+v, n-k], {v, 0, k}]; Table[t[n, k], {n, 2, 15}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Dec 20 2013 *)

Formula

T(n, k) = Sum_v C(n-k+1, 2*k-n-v)*C(n-k+v, n-k).

G.f. (1+x*y)^2/(1-x*y)*1/((1-x*y)-(1+x*y)*x^2*y). - Christopher Hanusa (chanusa(AT)math.washington.edu), Sep 22 2004

T(n,k) = 0 for n > 2*k. - Andrew Howroyd, Oct 02 2017

Extensions

More terms from James Sellers, Feb 06 2001