A324221 - OEIS

%I #20 Mar 18 2020 14:00:27

%S 0,1,6,15,36,72,139,244,414,663,1030,1540,2247,3187,4433,6036,8088,

%T 10658,13861,17785,22571,28329,35227,43401,53049,64333,77485,92697,

%U 110235,130324,153268,179326,208843,242115,279529,321422,368226,420319,478182,542238,613017

%N Number of connected 2n-regular loopless multigraphs with five nodes.

%C There are no (2n+1)-regular multigraphs satisfying the condition above.

%C Multigraphs are loopless.

%C Initial terms computed with 'Nauty and Traces'.

%H Brendan McKay and Adolfo Piperno, <a href="http://pallini.di.uniroma1.it">Nauty and Traces</a>

%F Conjectures from _Colin Barker_, Feb 18 2019: (Start)

%F G.f.: x*(1 + 3*x - x^2 + 4*x^3 - x^4 + 6*x^5 + 4*x^7 - x^8 - x^9 + x^10) / ((1 - x)^6*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).

%F a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) - 2*a(n-5) + 4*a(n-6) - 2*a(n-7) + a(n-8) - a(n-9) - 2*a(n-10) + 3*a(n-11) - a(n-12) for n>11.

%F (End)

%F Equivalent conjecture: 1152*a(n) = 6*n^5 + 30*n^4 + 220*n^3 + 540*n^2 + 1143*n - 353 + 72*A056594(n) + 128*A049347(n) + 153*A181983(n+1). - _R. J. Mathar_, Mar 09 2019

%o (nauty/shell) for ((n=0;n<76;n=n+2)); do geng -c -d1 5 -q | multig -m${n} -u; done

%Y Row n=5 of A328682.

%K nonn

%O 0,3

%A _Natan Arie Consigli_, Feb 18 2019

%E a(28)-a(30) from _Andrew Howroyd_, Mar 18 2020