A333467 Array read by antidiagonals: T(n,k) is the number of k-regular multigraphs on n labeled nodes, loops allowed, n >= 0, k >= 0.

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 2, 5, 3, 1, 1, 0, 3, 0, 17, 0, 1, 1, 1, 3, 15, 47, 73, 15, 1, 1, 0, 4, 0, 138, 0, 388, 0, 1, 1, 1, 4, 34, 306, 2021, 4720, 2461, 105, 1, 1, 0, 5, 0, 670, 0, 43581, 0, 18155, 0, 1, 1, 1, 5, 65, 1270, 25050, 291001, 1295493, 1256395, 152531, 945, 1

Offset

0,13

Links

Andrew Howroyd, Table of n, a(n) for n = 0..405 (diagonals 0..27)

Example

Array begins:
=============================================================
n\k | 0   1     2       3        4          5           6
----+--------------------------------------------------------
  0 | 1   1     1       1        1          1           1 ...
  1 | 1   0     1       0        1          0           1 ...
  2 | 1   1     2       2        3          3           4 ...
  3 | 1   0     5       0       15          0          34 ...
  4 | 1   3    17      47      138        306         670 ...
  5 | 1   0    73       0     2021          0       25050 ...
  6 | 1  15   388    4720    43581     291001     1594340 ...
  7 | 1   0  2461       0  1295493          0   159207201 ...
  8 | 1 105 18155 1256395 50752145 1296334697 23544232991 ...
  ...

Maple

b:= proc(l, i) option remember; (n-> `if`(n=0, 1,
     `if`(l[n]=0, b(sort(subsop(n=[][], l)), n-1),
     `if`(i<1, 0, b(l, i-1)+`if`(i=n, `if`(l[n]>1,
      b(subsop(n=l[n]-2, l), i), 0), `if`(l[i]>0,
      b(subsop(i=l[i]-1, n=l[n]-1, l), i), 0))))))(nops(l))
    end:
A:= (n, k)-> b([k$n], n):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 23 2020

Mathematica

b[l_, i_] := b[l, i] = Function[n, If[n == 0, 1, If[l[[n]] == 0, b[Sort[ ReplacePart[l, n -> Nothing]], n-1], If[i < 1, 0, b[l, i-1] + If[i == n, If[l[[n]] > 1, b[ReplacePart[l, n -> l[[n]]-2], i], 0], If[l[[i]] > 0, b[ReplacePart[l, {i -> l[[i]]-1, n -> l[[n]]-1}], i], 0]]]]]][Length[l]];
A[n_, k_] := b[Table[k, {n}], n];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 07 2020, after Alois P. Heinz *)

Prog

(PARI)
MultigraphsWLByDegreeSeq(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], forstep(e=0, limit, 2, recurse(n-r, limit, src[i, 1], 0, src[i, 2], e)))); Mat(M);
}
T(n, k)={if(n%2&&k%2, 0, vecsum(MultigraphsWLByDegreeSeq(n, k, (p, r)->subst(deriv(p), x, 1)>=(n-2*r)*k)[, 2]))}
{ for(n=0, 8, for(k=0, 6, print1(T(n, k), ", ")); print) }

Crossrefs

Rows n=0..3 are A000012, A059841, A008619, A006003.

Columns k=0..4 are A000012, A123023, A002135, A005814, A005816.

Cf. A059441 (graphs), A167625 (unlabeled nodes), A333351 (without loops).

Sequence in context: A166396 A152221 A144092 * A120648 A215401 A254606

Adjacent sequences: A333464 A333465 A333466 * A333468 A333469 A333470

Keyword

nonn,tabl

Author

Andrew Howroyd, Mar 23 2020

Status

approved