A327728 Number of unlabeled multigraphs with loops allowed and n edges covering three vertices.

0, 2, 8, 19, 40, 77, 132, 217, 340, 510, 742, 1054, 1456, 1976, 2634, 3453, 4464, 5703, 7194, 8987, 11120, 13636, 16588, 20036, 24024, 28630, 33916, 39951, 46816, 54601, 63376, 73253, 84324, 96690, 110466, 125778, 142728, 161468, 182126, 204841, 229768, 257075, 286902, 319447, 354880, 393384

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-3,0,6,0,-3,-2,1,2,-1).

Formula

a(n) = A050531(n) - A002620(n+2).

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - 3*a(n-4) + 6*a(n-6) - 3*a(n-8) - 2*a(n-9) + a(n-10) + 2*a(n-11) - a(n-12) for n > 12.

G.f.: x^2*(2 + 4*x + x^2 - 2*x^3 + x^6)/((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2).

Example

a(2) = 2 since three vertices may be covered with two edges in 2 ways: the path graph P(3) or an edge plus a loop.

Prog

(PARI) concat([0], Vec((2 + 4*x + x^2 - 2*x^3 + x^6)/((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2) + O(x^40)))

Crossrefs

Column k=3 of A327615.

Cf. A002620, A050531.

Sequence in context: A193389 A030504 A240279 * A000158 A101427 A286269

Adjacent sequences: A327725 A327726 A327727 * A327729 A327730 A327731

Keyword

nonn,easy

Author

Andrew Howroyd, Oct 23 2019

Status

approved