A391671 Number of different pawn structures on an n X n chessboard.

1, 1, 1, 27, 5407, 4342243, 6901005747, 15991700209747, 49095495585283107, 191599412918800541319, 926177959914373058950183, 5437262155990618367824723855, 38143289921919666735663285369003, 315385817450046831082107268064273565, 3037445494595218979993185134656710043205

Offset

0,4

Comments

On an n X n chessboard, a pawn structure is defined as the distribution of at most n white and n black pawns, subject to the constraint that the first and n-th ranks of the board remain free of pawns.

Links

Table of n, a(n) for n=0..14.

Formula

a(n) = Sum_{w=0..n} Sum_{b=0..n} binomial(n*(n-2),w) * binomial(n*(n-2)-w,b).

a(n) ~ exp(2*n-6) * n^(2*n-1) / (2*Pi). - Vaclav Kotesovec, Feb 12 2026

Example

For n = 3. Only row 2 with 3 squares needs to be considered. These squares can be empty (0), or hold a white pawn (w) or a black pawn (b). Structures: 000,00w,00b,0w0,0ww .. bbb. So, there are 3^3 = 27 structures possible.

Mathematica

a[n_]:=Sum[Sum[Binomial[n*(n-2), w]*Binomial[n*(n-2)-w, b], {b, 0, n}], {w, 0, n}]; Array[a, 15, 0] (* Stefano Spezia, Feb 12 2026 *)

Prog

(PARI) a(n) = sum(w=0, n, sum(b=0, n, binomial(n*(n-2), w) * binomial(n*(n-2)-w, b)));

Crossrefs

Cf. A384759.

Sequence in context: A211927 A298595 A278686 * A046367 A059795 A211928

Adjacent sequences: A391668 A391669 A391670 * A391672 A391673 A391674

Keyword

nonn

Author

Frank Hollstein, Feb 11 2026

Status

approved