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%I #43 Jun 06 2021 11:54:45
%S 1,2,8,58,632,9654,202484,5843954,233064944,12916716526,998745087980,
%T 108135391731690,16434082400952296,3513344943520006118,
%U 1058030578581541945316,449389062270642095128546,269419653009366144571801568,228157953744571034350576205790
%N Number of labeled split graphs on n vertices.
%C A split graph is a graph whose vertices can be partitioned into a clique and an independent set. - _Justin M. Troyka_, Oct 28 2018
%D 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).
%H Andrew Howroyd, <a href="/A179534/b179534.txt">Table of n, a(n) for n = 1..100</a>
%H E. A. Bender, L. B. Richmond, and N. C. Wormald, <a href="https://doi.org/10.1017/S1446788700023077">Almost all chordal graphs split</a>, J. Austral. Math. Soc. (Series A), 38 (1985), 214-221.
%H V. Bina, <a href="https://insis.vse.cz/zp/index.pl?prehled=vyhledavani;podrobnosti_zp=32066;zp=32066;dinfo_jazyk=3">Multidimensional probability distributions: Structure and learning</a>, PHD Thesis. Fac. of Management, University of Economics in Prague (2011).
%H J. M. Troyka, <a href="https://arxiv.org/abs/1803.07248">Split graphs: combinatorial species and asymptotics</a>, arXiv:1803.07248 [math.CO], 2018-2019.
%H J. M. Troyka, <a href="https://doi.org/10.37236/8278">Split graphs: combinatorial species and asymptotics</a>, Electron. J. Combin., 26 (2019), #P2.42.
%F 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) ).
%F From _Justin M. Troyka_, Oct 28 2018: (Start)
%F 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).
%F a(n) = A047863(n) - n*A047863(n-1) (see the Troyka link, Cor. 3.4).
%F a(n) ~ A047863(n) (see Bender, Richmond, and Wormald, Cor. 1). (End)
%p 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
%t 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 *)
%o (PARI)
%o b(n) = {sum(k=0, n, binomial(n, k)*2^(k*(n-k)))} \\ A047863(n)
%o a(n) = b(n) - n*b(n-1) \\ _Andrew Howroyd_, Jun 06 2021
%Y Cf. A047863, A048194, A058862, A006125.
%K nonn
%O 1,2
%A _Vladislav Bina_, Jul 18 2010
%E a(12)-a(16) corrected and terms a(17) and beyond from _Andrew Howroyd_, Jun 06 2021
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