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

Author

Andrew Howroyd, Oct 23 2019

Keywords

• nonn
• easy

Examples

			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.
		

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).

Crossrefs

Column k=3 of A327615.

Cf. A002620, A050531.

Programs

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)))

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).