A147878 The number of degree sequences with degree sum 2n representable by a connected graph (with multiple edges allowed).

1, 2, 5, 11, 23, 46, 86, 156, 273, 463, 766, 1241, 1969, 3073, 4723, 7157, 10711, 15850, 23206, 33654, 48373, 68955, 97544, 137002, 191125, 264955, 365127, 500349, 682018, 924982, 1248502, 1677530, 2244229, 2989952, 3967732, 5245354, 6909211

Offset

1,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

O. J. Rodseth, J. A. Sellers and H. Tverberg, Enumeration of the Degree Sequences of Non-Separable Graphs and Connected Graphs, European Journal of Combinatorics 30 (2009), 1301-1317.

Gus Wiseman, Connected multigraphs realizing each of the a(5) = 23 connected multigraphical graphical partitions of 10.

Formula

a(n) = p(2n) - p(n-1) - 2*Sum_{j=0..n-2} p(j).

a(n) = A000041(2*n) - 2*A000070(n) + 2*A000041(n) + A000041(n-1). - Vaclav Kotesovec, Nov 05 2016

a(n) ~ exp(2*Pi*sqrt(n/3))/(8*sqrt(3)*n) * (1 - (sqrt(3)/(2*Pi) + Pi/(48*sqrt(3))) /sqrt(n)). - Vaclav Kotesovec, Nov 05 2016

Example

From Gus Wiseman, Oct 26 2018: (Start)
The a(1) = 1 through a(5) = 23 connected multigraphical partitions:
  (11)  (22)   (33)    (44)     (55)
        (211)  (222)   (332)    (433)
               (321)   (422)    (442)
               (2211)  (431)    (532)
               (3111)  (2222)   (541)
                       (3221)   (3322)
                       (3311)   (3331)
                       (4211)   (4222)
                       (22211)  (4321)
                       (32111)  (4411)
                       (41111)  (5221)
                                (5311)
                                (22222)
                                (32221)
                                (33211)
                                (42211)
                                (43111)
                                (52111)
                                (222211)
                                (322111)
                                (331111)
                                (421111)
                                (511111)
(End)

Maple

with(combinat): seq(numbpart(2*m) - numbpart(m - 1) - 2*add(numbpart(j), j = 0 .. m-2), m=1..60);

Prog

(PARI) a(n) = numbpart(2*n) - numbpart(n-1) - 2*sum(j=0, n-2, numbpart(j)); \\ Michel Marcus, Nov 04 2016

Crossrefs

Cf. A000070, A000569, A007717, A096373, A147877, A209816, A320911, A320921 (no multiedges), A320923.

Sequence in context: A005986 A333396 A277828 * A179902 A140992 A248646

Adjacent sequences: A147875 A147876 A147877 * A147879 A147880 A147881

Keyword

nonn

Author

James A. Sellers, Nov 16 2008

Extensions

Offset corrected by Michel Marcus, Nov 04 2016

Status

approved