A179534 Number of labeled split graphs on n vertices.

1, 2, 8, 58, 632, 9654, 202484, 5843954, 233064944, 12916716526, 998745087980, 108135391731690, 16434082400952296, 3513344943520006118, 1058030578581541945316, 449389062270642095128546, 269419653009366144571801568, 228157953744571034350576205790

Offset

1,2

Comments

A split graph is a graph whose vertices can be partitioned into a clique and an independent set. - Justin M. Troyka, Oct 28 2018

References

V. Bina, Enumeration of Labeled Split Graphs and Counts of Important Superclasses. In Adacher L., Flamini, M., Leo, G., Nicosia, G., Pacifici, A., Picialli, V. (Eds.). CTW 2011, Villa Mondragone, Frascati, pp. 72-75 (2011).

Links

Andrew Howroyd, Table of n, a(n) for n = 1..100

E. A. Bender, L. B. Richmond, and N. C. Wormald, Almost all chordal graphs split, J. Austral. Math. Soc. (Series A), 38 (1985), 214-221.

V. Bina, Multidimensional probability distributions: Structure and learning, PHD Thesis. Fac. of Management, University of Economics in Prague (2011).

J. M. Troyka, Split graphs: combinatorial species and asymptotics, arXiv:1803.07248 [math.CO], 2018-2019.

J. M. Troyka, Split graphs: combinatorial species and asymptotics, Electron. J. Combin., 26 (2019), #P2.42.

Formula

a(n) = 1 + Sum_{k=2..n} binomial(n,k) ( (2^k-1)^(n-k) - Sum_{j=1..n-k) j*k*binomial(n-k,j)*(2^(k-1)-1)^(n-k-j) /(j+1) ).

From Justin M. Troyka, Oct 28 2018: (Start)

a(n) = [ Sum_{k=0..n} binomial(n,k) 2^(k(n-k)) ] - [ n Sum_{k=0..n-1} binomial(n-1,k)*2^(k(n-k-1)) ] (see the Troyka link, Cor. 3.4).

a(n) = A047863(n) - n*A047863(n-1) (see the Troyka link, Cor. 3.4).

a(n) ~ A047863(n) (see Bender, Richmond, and Wormald, Cor. 1). (End)

Maple

A179534 := proc(n) local a, k; a := 1 ; for k from 2 to n do a := a+binomial(n, k)*( (2^k-1)^(n-k) -add(j*k*binomial(n-k, j)*(2^(k-1)-1)^(n-k-j)/(j+1), j=1..n-k) ) end do: a ; end proc: # R. J. Mathar, Jun 21 2011

Mathematica

a[n_] := 1 + Sum[Binomial[n, k]*((2^k-1)^(n-k) - Sum[j*k*Binomial[n-k, j]*(2^(k-1)-1)^(n-k-j)/(j+1), {j, 1, n-k}]), {k, 2, n}]; Array[a, 20] (* Stefano Spezia, Oct 29 2018 *)

Prog

(PARI)
b(n) = {sum(k=0, n, binomial(n, k)*2^(k*(n-k)))} \\ A047863(n)
a(n) = b(n) - n*b(n-1) \\ Andrew Howroyd, Jun 06 2021

Crossrefs

Cf. A047863, A048194, A058862, A006125.

Sequence in context: A319590 A005804 A162067 * A256034 A086907 A132186

Adjacent sequences: A179531 A179532 A179533 * A179535 A179536 A179537

Keyword

nonn

Author

Vladislav Bina, Jul 18 2010

Extensions

a(12)-a(16) corrected and terms a(17) and beyond from Andrew Howroyd, Jun 06 2021

Status

approved