A059963 Triangle T(n,k) giving number of ways of placing n nonattacking queens on n X n board with the queen on the first row fixed at column k, 1<=k<=n.

1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 2, 2, 0, 1, 1, 1, 1, 0, 4, 7, 6, 6, 6, 7, 4, 4, 8, 16, 18, 18, 16, 8, 4, 28, 30, 47, 44, 54, 44, 47, 30, 28, 64, 48, 65, 93, 92, 92, 93, 65, 48, 64, 96, 219, 209, 295, 346, 350, 346, 295, 209, 219, 96, 500, 806, 1165, 1359, 1631, 1639

Offset: 1

Author

Yong Kong (ykong(AT)curagen.com), Mar 03 2001

Keywords

• nonn
• tabl

Comments

A000170 (non-attacking queens) can be derived from this sequence as follows: a(12)= 2*(S1(12)+S2(12)+S3(12)+S4(12)+S5(12)+S6(12)) when n is even, a(13)=S7(13) + 2*(S1(13)+S2(13)+S3(13)+S4(13)+S5(13)+S6(13)) when n is odd. Here Si(j) means T(j,i). - Patrick R. GUILLEMIN (patrick.guillemin(AT)etsi.org), Jan 05 2004

Examples

			When n = 8 there are 16 ways to place if the queen on the first row is at the third column
Triangle begins:
1,
0,0,
0,0,0,
0,1,1,0,
2,2,2,2,2,
0,1,1,1,1,0,
4,7,6,6,6,7,4,
4,8,16,18,18,16,8,4,
28,30,47,44,54,44,47,30,28, etc.
		

Links

• Patrick R. GUILLEMIN, Extension of triangle to 22 rows
• Patrick R. GUILLEMIN, Extension of triangle to 22 rows

Crossrefs

Cf. A000170.

Extensions

Confirmed by Patrick R. GUILLEMIN (patrick.guillemin(AT)etsi.org), who, together with colleagues, has computed the first 21 rows of this triangle, Jan 05 2004

Sep 15 2004: Patrick R. GUILLEMIN (patrick.guillemin(AT)etsi.org), together with colleagues, has computed the 22nd row of this triangle.