A361749 a(n) is the number of n X n matrices with nonnegative integer entries, row sums 1,2,...,n and column sums 1,2,...,n.

1, 1, 2, 12, 261, 22645, 8264346, 13150070522, 93589674933872, 3036609755945925595, 455845471095088280120142, 320342093420041869298750385976, 1063978124653925432733949863518874116, 16835366182312565093823092118182447742597067

Offset: 0

Author

Robert Israel, Mar 23 2023

Keywords

• nonn

Examples

			a(3) = 12 because there are 12 possible 3 X 3 matrices with nonnegative integer entries, row sums 1,2,3 and column sums 1,2,3:
  [ 0 0 1 ]   [ 0 0 1 ]   [ 0 0 1 ]   [ 0 0 1 ]
  [ 0 0 2 ]   [ 0 1 1 ]   [ 0 2 0 ]   [ 1 0 1 ]
  [ 1 2 0 ],  [ 1 1 1 ],  [ 1 0 2 ],  [ 0 2 1 ],
  .
  [ 0 0 1 ]   [ 0 1 0 ]   [ 0 1 0 ]   [ 0 1 0 ]
  [ 1 1 0 ]   [ 0 0 2 ]   [ 0 1 1 ]   [ 1 0 1 ]
  [ 0 1 2 ],  [ 1 1 1 ],  [ 1 0 2 ],  [ 0 1 2 ],
  .
  [ 0 1 0 ]   [ 1 0 0 ]   [ 1 0 0 ]   [ 1 0 0 ]
  [ 1 1 0 ]   [ 0 0 2 ]   [ 0 1 1 ]   [ 0 2 0 ]
  [ 0 0 3 ],  [ 0 2 1 ],  [ 0 1 2 ],  [ 0 0 3 ].
		

Links

• Andrew Howroyd, PARI Program, Mar 2023.

Crossrefs

Cf. A000681, A110058.

Programs

Maple

G:= proc(L,R,k) option remember;
# number of solutions with first k entries of first row 0
local m,n,i;
m:= nops(L); n:= nops(R);
if m <= 1 then return 1 fi;
if L[1] > convert(R[k+1..n],`+`) then return 0 fi;
if k = n-1 then return procname(L[2..-1],subsop(n = R[n]-L[1], R),0) fi;
add(procname(subsop(1=L[1]-i, L), subsop(k+1=R[k+1]-i, R), k+1), i=0..min(L[1],R[k+1]))
end proc:
seq(G([$1..n],[$1..n],0), n=0..8);

Extensions

a(10)-a(13) from Andrew Howroyd, Mar 29 2023

a(0)=1 prepended by Alois P. Heinz, Jun 26 2023