This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A179534 #50 Dec 20 2024 23:39:35
%S A179534 1,2,8,58,632,9654,202484,5843954,233064944,12916716526,998745087980,
%T A179534 108135391731690,16434082400952296,3513344943520006118,
%U A179534 1058030578581541945316,449389062270642095128546,269419653009366144571801568,228157953744571034350576205790
%N A179534 Number of labeled split graphs on n vertices.
%C A179534 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 A179534 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 A179534 Andrew Howroyd, <a href="/A179534/b179534.txt">Table of n, a(n) for n = 1..100</a>
%H A179534 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 A179534 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 A179534 Venkatesan Guruswami, <a href="https://doi.org/10.1016/S0012-365X(99)00022-9">Enumerative aspects of certain subclasses of perfect graphs</a>, Discrete Mathematics 205 (1999), 97-117.
%H A179534 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 A179534 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 A179534 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 A179534 From _Justin M. Troyka_, Oct 28 2018: (Start)
%F A179534 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 A179534 a(n) = A047863(n) - n*A047863(n-1) (see the Troyka link, Cor. 3.4).
%F A179534 a(n) ~ A047863(n) (see Bender, Richmond, and Wormald, Cor. 1). (End)
%p A179534 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 A179534 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 A179534 (PARI)
%o A179534 b(n) = {sum(k=0, n, binomial(n, k)*2^(k*(n-k)))} \\ A047863(n)
%o A179534 a(n) = b(n) - n*b(n-1) \\ _Andrew Howroyd_, Jun 06 2021
%Y A179534 Cf. A047863, A048194, A058862, A006125.
%K A179534 nonn,easy
%O A179534 1,2
%A A179534 _Vladislav Bina_, Jul 18 2010
%E A179534 a(12)-a(16) corrected and terms a(17) and beyond from _Andrew Howroyd_, Jun 06 2021