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A006129 a(0), a(1), a(2), ... satisfy Sum_{k=0..n} a(k)*binomial(n,k) = 2^binomial(n,2), for n >= 0.

%I A006129 M3678 #74 Feb 16 2025 08:32:29
%S A006129 1,0,1,4,41,768,27449,1887284,252522481,66376424160,34509011894545,
%T A006129 35645504882731588,73356937912127722841,301275024444053951967648,
%U A006129 2471655539737552842139838345,40527712706903544101000417059892,1328579255614092968399503598175745633
%N A006129 a(0), a(1), a(2), ... satisfy Sum_{k=0..n} a(k)*binomial(n,k) = 2^binomial(n,2), for n >= 0.
%C A006129 Also labeled graphs on n unisolated nodes (inverse binomial transform of A006125). - _Vladeta Jovovic_, Apr 09 2000
%C A006129 Also the number of edge covers of the complete graph K_n. - _Eric W. Weisstein_, Mar 30 2017
%D A006129 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006129 Alois P. Heinz, <a href="/A006129/b006129.txt">Table of n, a(n) for n = 0..50</a>
%H A006129 A. N. Bhavale, B. N. Waphare, <a href="https://ajc.maths.uq.edu.au/pdf/78/ajc_v78_p073.pdf">Basic retracts and counting of lattices</a>, Australasian J. of Combinatorics (2020) Vol. 78, No. 1, 73-99.
%H A006129 C. L. Mallows & N. J. A. Sloane, <a href="/A006123/a006123.pdf">Emails, May 1991</a>
%H A006129 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A006129 R. Tauraso, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Tauraso/tauraso18.html">Edge cover time for regular graphs</a>, JIS 11 (2008) 08.4.4.
%H A006129 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteGraph.html">Complete Graph</a>
%H A006129 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EdgeCover.html">Edge Cover</a>
%F A006129 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*2^binomial(k, 2).
%F A006129 E.g.f.: A(x)/exp(x) where A(x) = Sum_{n>=0} 2^C(n,2) x^n/n!. - _Geoffrey Critzer_, Oct 21 2011
%F A006129 a(n) ~ 2^(n*(n-1)/2). - _Vaclav Kotesovec_, May 04 2015
%e A006129 2^binomial(n,2) = 1 + binomial(n,2) + 4*binomial(n,3) + 41*binomial(n,4) + 768*binomial(n,5) + ...
%p A006129 a:= proc(n) option remember; `if`(n=0, 1,
%p A006129       2^binomial(n, 2) - add(a(k)*binomial(n,k), k=0..n-1))
%p A006129     end:
%p A006129 seq(a(n), n=0..20);  # _Alois P. Heinz_, Oct 26 2012
%t A006129 a = Sum[2^Binomial[n, 2] x^n/n!, {n, 0, 20}]; Range[0, 20]! CoefficientList[Series[a/Exp[x], {x, 0, 20}], x] (* _Geoffrey Critzer_, Oct 21 2011 *)
%t A006129 Table[Sum[(-1)^(n - k) Binomial[n, k] 2^Binomial[k, 2], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, May 04 2015 *)
%o A006129 (PARI) for(n=0,25, print1(sum(k=0,n,(-1)^(n-k)*binomial(n, k)*2^binomial(k, 2)), ", ")) \\ _G. C. Greubel_, Mar 30 2017
%o A006129 (Python)
%o A006129 from sympy.core.cache import cacheit
%o A006129 from sympy import binomial
%o A006129 @cacheit
%o A006129 def a(n): return 1 if n==0 else 2**binomial(n, 2) - sum(a(k)*binomial(n, k) for k in range(n))
%o A006129 print([a(n) for n in range(21)]) # _Indranil Ghosh_, Aug 12 2017
%Y A006129 Row sums of A054548.
%Y A006129 Cf. A322661 (if loops allowed), A086193 (directed edges), A002494 (unlabeled).
%K A006129 nonn,nice,easy
%O A006129 0,4
%A A006129 _Colin Mallows_
%E A006129 More terms from _Vladeta Jovovic_, Apr 09 2000