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A333351
Array read by antidiagonals: T(n,k) is the number of k-regular loopless multigraphs on n labeled nodes, n >= 0, k >= 0.
9
1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 3, 1, 1, 0, 1, 0, 6, 0, 1, 1, 0, 1, 1, 10, 22, 15, 1, 1, 0, 1, 0, 15, 0, 130, 0, 1, 1, 0, 1, 1, 21, 158, 760, 822, 105, 1, 1, 0, 1, 0, 28, 0, 3355, 0, 6202, 0, 1, 1, 0, 1, 1, 36, 654, 12043, 93708, 190050, 52552, 945, 1, 1, 0, 1, 0, 45, 0, 36935, 0, 3535448, 0, 499194, 0, 1
OFFSET
0,20
LINKS
EXAMPLE
Array begins:
=================================================================
n\k | 0 1 2 3 4 5 6 7
----+------------------------------------------------------------
0 | 1 1 1 1 1 1 1 1 ...
1 | 1 0 0 0 0 0 0 0 ...
2 | 1 1 1 1 1 1 1 1 ...
3 | 1 0 1 0 1 0 1 0 ...
4 | 1 3 6 10 15 21 28 36 ...
5 | 1 0 22 0 158 0 654 0 ...
6 | 1 15 130 760 3355 12043 36935 100135 ...
7 | 1 0 822 0 93708 0 3226107 0 ...
8 | 1 105 6202 190050 3535448 45163496 431400774 3270643750 ...
...
PROG
(PARI)
MultigraphsByDegreeSeq(n, limit, ok)={
local(M=Map(Mat([0, 1])));
my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
my(recurse(r, h, p, q, v, e) = if(!p, if(ok(x^e+q, r), acc(x^e+q, v)), my(i=poldegree(p), t=pollead(p)); self()(r, limit, p-t*x^i, q+t*x^i, v, e); for(m=1, h-i, for(k=1, min(t, (limit-e)\m), self()(r, if(k==t, limit, i+m-1), p-k*x^i, q+k*x^(i+m), binomial(t, k)*v, e+k*m)))));
for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(n-r, limit, src[i, 1], 0, src[i, 2], 0))); Mat(M);
}
T(n, k)={if((n%2&&k%2)||(n==1&&k>0), 0, vecsum(MultigraphsByDegreeSeq(n, k, (p, r)->subst(deriv(p), x, 1)>=(n-2*r)*k)[, 2]))}
{ for(n=0, 8, for(k=0, 7, print1(T(n, k), ", ")); print) }
CROSSREFS
Rows n=4..6 are A000217(n+1), A244868 (with interspersed zeros), A244878.
Columns k=0..4 are A000012, A123023, A002137, A108243 (with interspersed zeros), A367497.
Cf. A059441 (graphs), A333157, A333330 (unlabeled nodes), A333467 (with loops).
Sequence in context: A195151 A271024 A125208 * A193140 A319993 A122776
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Mar 15 2020
STATUS
approved