A390167 Array read by antidiagonals: T(n,k) is the number of edge-weighted k-regular multigraphs on 2 unlabeled vertices with total edge weight n.

1, 1, 0, 2, 1, 0, 2, 2, 1, 0, 3, 3, 4, 1, 0, 3, 4, 6, 4, 1, 0, 4, 5, 10, 9, 6, 1, 0, 4, 6, 12, 15, 13, 6, 1, 0, 5, 7, 16, 21, 27, 17, 8, 1, 0, 5, 8, 18, 28, 38, 36, 23, 8, 1, 0, 6, 9, 22, 34, 55, 59, 55, 28, 10, 1, 0, 6, 10, 24, 41, 67, 87, 92, 71, 35, 10, 1, 0

Offset

0,4

Comments

Loops are allowed and add two to the degree of the vertex. Since the graph is regular and there are exactly two vertices, the number of loops at both of the vertices will be equal.

For k > 2*n, only edges of weight 0 can be added. Hence T(n,k) - T(n,k-2) becomes constant. This explains why all rows satisfy the recurrence T(n,k) = T(n,k-1) + T(n,k-2) - T(n,k-3) for k > 2*n.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1274 (first 51 antidiagonals)

Formula

G.f. of column k: Sum_{j=0..floor(k/2)} (P(j,x)^2 + P(j,x^2))*P(k-2*j,x)/2, where P(k,x) = 1/Product_{i=1..k} (1 - x^i).

T(n,k) = T(n,k-1) + T(n,k-2) - T(n,k-3) for n > 0, k > 2*n.

Example

Array begins:
=============================================
n\k | 0 1  2  3   4   5   6   7    8    9 ...
----+----------------------------------------
  0 | 1 1  2  2   3   3   4   4    5    5 ...
  1 | 0 1  2  3   4   5   6   7    8    9 ...
  2 | 0 1  4  6  10  12  16  18   22   24 ...
  3 | 0 1  4  9  15  21  28  34   41   47 ...
  4 | 0 1  6 13  27  38  55  67   85   97 ...
  5 | 0 1  6 17  36  59  87 114  145  173 ...
  6 | 0 1  8 23  55  92 147 196  260  313 ...
  7 | 0 1  8 28  71 132 217 309  417  521 ...
  8 | 0 1 10 35  99 190 332 484  680  863 ...
  9 | 0 1 10 42 123 259 469 721 1034 1359 ...
  ...
See the examples in A358244 for T(4,2). Most of the other row sequences also give an example.

Prog

(PARI)
P(n, k) = { 1/prod(i=1, min(n, k), 1 - x^i, 1 + O(x*x^n)) }
T(n, k) = { polcoef(sum(j=0, k\2, (P(n, j)^2 + subst(P(n\2, j), x, x^2))*P(n, k-2*j)), n)/2 }

Crossrefs

Rows 0..9 are A008619, A001477, A047233(n>1), A358243, A358244, A358245, A358246, A358247, A358248, A358249.

Main diagonal is A390166.

Sequence in context: A289281 A374996 A212957 * A035393 A068913 A128306

Adjacent sequences: A390164 A390165 A390166 * A390168 A390169 A390170

Keyword

nonn,tabl

Author

Andrew Howroyd, Nov 22 2025

Status

approved