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) (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
KEYWORD
nonn,easy,changed
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