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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. (Formerly M3678) 60
 1, 0, 1, 4, 41, 768, 27449, 1887284, 252522481, 66376424160, 34509011894545, 35645504882731588, 73356937912127722841, 301275024444053951967648, 2471655539737552842139838345, 40527712706903544101000417059892, 1328579255614092968399503598175745633 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Also labeled graphs on n unisolated nodes (inverse binomial transform of A006125). - Vladeta Jovovic, Apr 09 2000 Also the number of edge covers of the complete graph K_n. - Eric W. Weisstein, Mar 30 2017 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..50 A. N. Bhavale, B. N. Waphare, Basic retracts and counting of lattices, Australasian J. of Combinatorics (2020) Vol. 78, No. 1, 73-99. C. L. Mallows & N. J. A. Sloane, Emails, May 1991 N. J. A. Sloane, Transforms R. Tauraso, Edge cover time for regular graphs, JIS 11 (2008) 08.4.4. Eric Weisstein's World of Mathematics, Complete Graph Eric Weisstein's World of Mathematics, Edge Cover FORMULA a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*2^binomial(k, 2). 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 a(n) ~ 2^(n*(n-1)/2). - Vaclav Kotesovec, May 04 2015 EXAMPLE 2^binomial(n,2) = 1 + binomial(n,2) + 4*binomial(n,3) + 41*binomial(n,4) + 768*binomial(n,5) + ... MAPLE a:= proc(n) option remember; `if`(n=0, 1,       2^binomial(n, 2) - add(a(k)*binomial(n, k), k=0..n-1))     end: seq(a(n), n=0..20);  # Alois P. Heinz, Oct 26 2012 MATHEMATICA 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 *) Table[Sum[(-1)^(n - k) Binomial[n, k] 2^Binomial[k, 2], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 04 2015 *) PROG (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 (Python) from sympy.core.cache import cacheit from sympy import binomial @cacheit def a(n): return 1 if n==0 else 2**binomial(n, 2) - sum(a(k)*binomial(n, k) for k in range(n)) print([a(n) for n in range(21)]) # Indranil Ghosh, Aug 12 2017 CROSSREFS Sequence in context: A001908 A270703 A192547 * A244437 A265003 A193363 Adjacent sequences:  A006126 A006127 A006128 * A006130 A006131 A006132 KEYWORD nonn,nice,easy,changed AUTHOR EXTENSIONS More terms from Vladeta Jovovic, Apr 09 2000 STATUS approved

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Last modified October 25 11:56 EDT 2020. Contains 338012 sequences. (Running on oeis4.)