A050532 Number of multigraphs with loops on 4 nodes with n edges.

1, 2, 7, 20, 53, 125, 287, 606, 1226, 2358, 4356, 7740, 13327, 22239, 36151, 57336, 88962, 135249, 201912, 296324, 428211, 609935, 857327, 1190216, 1633551, 2218011, 2981607, 3970548, 5241120, 6861024, 8911782, 11490282, 14711976, 18712911, 23653440

Offset: 0

Author

Vladeta Jovovic, Dec 29 1999

Keywords

• easy
• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-2,-2,-2,11,-1,1,-10,-10,12,4,12,-10,-10,1,-1,11,-2,-2,-2,-1,3,-1).

Crossrefs

Column k=4 of A290428.

Cf. A002620.

Programs

Mathematica

<< Combinatorica`
nn = 30; n = 4; CoefficientList[Series[CycleIndex[Join[PairGroup[SymmetricGroup[n]], Permutations[Range[n*(n - 1)/2 + 1, n*(n + 1)/2]], 2], s] /.Table[s[i] -> 1/(1 - x^i), {i, 1, n^2 - n}], {x, 0, nn}], x] (* Geoffrey Critzer, Aug 07 2015 *)
CoefficientList[Series[(4 x^11 + 4 x^10 + 11 x^9 + 15 x^8 + 9 x^7 + 12 x^6 + 6 x^5 + 6 x^4 + 3 x^3 + 2 x^2 - x + 1)/((x^4 - 1)^2 (x^3 - 1)^3 (x^2 - 1)^2 (x-1)^3), {x, 0, 35}], x] (* Vincenzo Librandi, Aug 08 2015 *)

PARI

Vec((1 - x + 2*x^2 + 3*x^3 + 6*x^4 + 6*x^5 + 12*x^6 + 9*x^7 + 15*x^8 + 11*x^9 + 4*x^10 + 4*x^11)/((1 - x)^3*(1 - x^2)^2*(1 - x^3)^3*(1 - x^4)^2) + O(x^40)) \\ Andrew Howroyd, Mar 16 2020

Formula

G.f.: (4*x^11+4*x^10+11*x^9+15*x^8+9*x^7+12*x^6+6*x^5+6*x^4+3*x^3+2*x^2-x+1)/((x^4-1)^2*(x^3-1)^3*(x^2-1)^2*(x-1)^3).

Extensions

a(33)-a(34) from Vincenzo Librandi, Aug 08 2015