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A284989 Triangle T(n,k) read by rows: the number of n X n {0,1} matrices with trace k where each row sum and each column sum is 2.

%I A284989 #35 Dec 24 2018 21:28:42
%S A284989 1,0,0,0,0,1,1,0,3,2,9,24,24,24,9,216,540,610,420,210,44,7570,18000,
%T A284989 20175,13720,6300,1920,265,357435,829920,909741,617610,284235,91140,
%U A284989 19005,1854,22040361,50223600,54295528,36663312,17072790,5679184,1337280,203952,14833
%N A284989 Triangle T(n,k) read by rows: the number of n X n {0,1} matrices with trace k where each row sum and each column sum is 2.
%H A284989 Alois P. Heinz, <a href="/A284989/b284989.txt">Rows n = 0..14, flattened</a>
%H A284989 <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>
%F A284989 Let z1..zn be n variables and s1 = Sum_{k=1..n} zk, s2 = Sum_{k=1..n} zk^2, s12 = (s1^2 - s2)/2, fk = t*(s12 - zk*(s1 - zk)) + zk*(s1 - zk) for k=1..n, P_n(t) = [(z1..zn)^2] Product_{k=1..n} fk. Then P_n(t) = Sum_{k=0..n} T(n,k)*t^(n-k), n >= 3. - _Gheorghe Coserea_, Dec 21 2018
%e A284989 0:         1
%e A284989 1:         0        0
%e A284989 2:         0        0        1
%e A284989 3:         1        0        3        2
%e A284989 4:         9       24       24       24        9
%e A284989 5:       216      540      610      420      210      44
%e A284989 6:      7570    18000    20175    13720     6300    1920     265
%e A284989 7:    357435   829920   909741   617610   284235   91140   19005   1854
%e A284989 8:  22040361 50223600 54295528 36663312 17072790 5679184 1337280 203952 14833
%o A284989 (PARI)
%o A284989 P(n, t='t) = {
%o A284989   my(z=vector(n, k, eval(Str("z", k))),
%o A284989      s1=sum(k=1, #z, z[k]), s2=sum(k=1, #z, z[k]^2), s12=(s1^2 - s2)/2,
%o A284989      f=vector(n, k, t*(s12 - z[k]*(s1 - z[k])) + z[k]*(s1 - z[k])), g=1);
%o A284989   for (i=1, n, g *= f[i]; for(j=1, n, g=substpol(g, z[j]^3, 0)));
%o A284989   for (k=1, n, g=polcoef(g, 2, z[k]));
%o A284989   g;
%o A284989 };
%o A284989 seq(N) = concat([[1], [0, 0], [0, 0, 1]], apply(n->Vec(P(n)), [3..N]));
%o A284989 concat(seq(8)) \\ _Gheorghe Coserea_, Dec 21 2018
%Y A284989 Cf. A001499 (row sums), A000166 (diagonal), A007107 (column 0).
%Y A284989 Cf. A008290, A098825, A284990, A284991.
%K A284989 nonn,tabl
%O A284989 0,9
%A A284989 _R. J. Mathar_, Apr 07 2017