A050929 - OEIS

%I #16 Mar 16 2020 20:13:48

%S 1,2,11,47,198,713,2423,7388,21003,55433,137944,324659,729022,1567139,

%T 3242954,6479759,12547894,23607614,43267994,77405064,135435666,

%U 232137202,390371944,644897542,1047890293,1676518363,2643628813

%N Number of directed multigraphs with loops on 4 nodes with n arcs.

%H Robert Israel, <a href="/A050929/b050929.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: (x^26-x^25 + 4*x^24 + 18*x^23 + 63*x^22 + 151*x^21 + 402*x^20 + 790*x^19 + 1511*x^18 + 2353*x^17 + 3400*x^16 + 4296*x^15 + 5115*x^14 + 5266*x^13 + 5115*x^12 + 4296*x^11 + 3400*x^10 + 2353*x^9 + 1511*x^8 + 790*x^7 + 402*x^6 + 151*x^5 + 63*x^4 + 18*x^3 + 4*x^2-x + 1)/((x^4-1)^4*(x^3-1)^5*(x^2-1)^4*(x-1)^3).

%p gf:= (x^26-x^25 + 4*x^24 + 18*x^23 + 63*x^22 + 151*x^21 + 402*x^20 + 790*x^19 + 1511*x^18 + 2353*x^17 + 3400*x^16 + 4296*x^15 + 5115*x^14 + 5266*x^13 + 5115*x^12 + 4296*x^11 + 3400*x^10 + 2353*x^9 + 1511*x^8 + 790*x^7 + 402*x^6 + 151*x^5 + 63*x^4 + 18*x^3 + 4*x^2-x + 1)/((x^4-1)^4*(x^3-1)^5*(x^2-1)^4*(x-1)^3):

%p S:= series(gf,x,101):

%p seq(coeff(S,x,j),j=0..100); # _Robert Israel_, Aug 07 2015

%t nn = 30; n = 4; CoefficientList[Series[CycleIndex[ Join[PairGroup[SymmetricGroup[n], Ordered], Permutations[Range[n*(n - 1) + 1, n*(n - 1) + n]], 2], s] /. Table[s[i] -> 1/(1 - x^i), {i, 1, n^2 - n}], {x, 0, nn}], x] (* _Geoffrey Critzer_, Aug 07 2015*)

%Y Column k=4 of A138107.

%Y Cf. A005993.

%K easy,nonn

%O 0,2

%A _Vladeta Jovovic_, Dec 30 1999