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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.

%I A361749 #18 Jun 26 2023 09:40:30
%S A361749 1,1,2,12,261,22645,8264346,13150070522,93589674933872,
%T A361749 3036609755945925595,455845471095088280120142,
%U A361749 320342093420041869298750385976,1063978124653925432733949863518874116,16835366182312565093823092118182447742597067
%N 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.
%H A361749 Andrew Howroyd, <a href="/A361749/a361749.gp.txt">PARI Program</a>, Mar 2023.
%e A361749 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:
%e A361749   [ 0 0 1 ]   [ 0 0 1 ]   [ 0 0 1 ]   [ 0 0 1 ]
%e A361749   [ 0 0 2 ]   [ 0 1 1 ]   [ 0 2 0 ]   [ 1 0 1 ]
%e A361749   [ 1 2 0 ],  [ 1 1 1 ],  [ 1 0 2 ],  [ 0 2 1 ],
%e A361749   .
%e A361749   [ 0 0 1 ]   [ 0 1 0 ]   [ 0 1 0 ]   [ 0 1 0 ]
%e A361749   [ 1 1 0 ]   [ 0 0 2 ]   [ 0 1 1 ]   [ 1 0 1 ]
%e A361749   [ 0 1 2 ],  [ 1 1 1 ],  [ 1 0 2 ],  [ 0 1 2 ],
%e A361749   .
%e A361749   [ 0 1 0 ]   [ 1 0 0 ]   [ 1 0 0 ]   [ 1 0 0 ]
%e A361749   [ 1 1 0 ]   [ 0 0 2 ]   [ 0 1 1 ]   [ 0 2 0 ]
%e A361749   [ 0 0 3 ],  [ 0 2 1 ],  [ 0 1 2 ],  [ 0 0 3 ].
%p A361749 G:= proc(L,R,k) option remember;
%p A361749 # number of solutions with first k entries of first row 0
%p A361749 local m,n,i;
%p A361749 m:= nops(L); n:= nops(R);
%p A361749 if m <= 1 then return 1 fi;
%p A361749 if L[1] > convert(R[k+1..n],`+`) then return 0 fi;
%p A361749 if k = n-1 then return procname(L[2..-1],subsop(n = R[n]-L[1], R),0) fi;
%p A361749 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]))
%p A361749 end proc:
%p A361749 seq(G([$1..n],[$1..n],0), n=0..8);
%Y A361749 Cf. A000681, A110058.
%K A361749 nonn
%O A361749 0,3
%A A361749 _Robert Israel_, Mar 23 2023
%E A361749 a(10)-a(13) from _Andrew Howroyd_, Mar 29 2023
%E A361749 a(0)=1 prepended by _Alois P. Heinz_, Jun 26 2023