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

0,3

Links

Table of n, a(n) for n=0..13.

Andrew Howroyd, PARI Program, Mar 2023.

Example

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 ].

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);

Crossrefs

Cf. A000681, A110058.

Sequence in context: A369680 A217652 A320212 * A012549 A009610 A012546

Adjacent sequences: A361746 A361747 A361748 * A361750 A361751 A361752

Keyword

nonn

Author

Robert Israel, Mar 23 2023

Extensions

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

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

Status

approved