

A130102


E.g.f.: (e^x  x)^2.


5



1, 0, 2, 2, 8, 22, 52, 114, 240, 494, 1004, 2026, 4072, 8166, 16356, 32738, 65504, 131038, 262108, 524250, 1048536, 2097110, 4194260, 8388562, 16777168, 33554382, 67108812, 134217674, 268435400, 536870854, 1073741764, 2147483586
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OFFSET

0,3


COMMENTS

a(n) is the number of length n binary sequences in which no symbol occurs exactly once. (The Rosenthal formula takes 2^n for the total number of binary sequences and subtracts n for each sequence of length n with a single 0 or 1.)  Geoffrey Critzer, Dec 03 2011
From Ambrosio ValenciaRomero, Mar 08 2022: (Start)
a(n), for n > 1, is the number of pure Nash equilibria in the symmetric nplayer twostrategy normalform unanimity game. Let i be a player in set N = {1, 2, 3, ..., n} and s(i) in set S = {0, 1} be i's strategy. Then i's payoff, u(i), in this game is given by:
u(i) = 1 if s(1) = s(2) = ... = s(n1) = s(n); otherwise, u(i) = 0.
Only two of the a(n) pure equilibria in this unanimity game are strict: s = <0, 0, ..., 0, 0> and s = <1, 1, ..., 1, 1>; these are the diagonal collective strategies where all actors obtain the payoff u(i) = 1.
The other a(n)2 pure equilibria are weak and produce an individual payoff of u(i) = 0; these correspond to the collective strategy outcomes where more than one and fewer than n1 individual strategies differ. For instance, for n = 4, the a(4)2 = 6 weak pure equilibria are <0, 0, 1, 1>, <0, 1, 0, 1>, <0, 1, 1, 0>, <1, 0, 0, 1>, <1, 0, 1, 0>, and <1, 1, 0, 0>. (End)


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,5,2).


FORMULA

a(n) = 2^n  2*n for n <> 2 (cf. A005803).  Rainer Rosenthal, Feb 14 2010.
E.g.f.: e^(2*x)  2*x*e^x + x^2.
a(n) = Sum_{k=0..n} binomial(n,k)*A060576(k)*A060576(nk).
G.f. 1 + 2*x^2  2*x^3/((2*x  1)*(x  1)^2).  R. J. Mathar, Dec 04 2011


EXAMPLE

a(4) = 8 because there are 8 sequences on {0,1} such that neither 0 nor 1 occurs exactly once: {0,0,0,0}, {0,0,1,1}, {0,1,0,1}, {0,1,1,0}, {1,0,0,1}, {1,0,1,0}, {1,1,0,0}, {1,1,1,1}.  Geoffrey Critzer, Dec 03 2011


MATHEMATICA

a=Exp[x]x; Range[0, 20]! CoefficientList[Series[a^2, {x, 0, 20}], x] (* Geoffrey Critzer, Dec 03 2011 *)
CoefficientList[Series[1+2*x^22*x^3/((2*x1)*(x1)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 28 2012 *)


PROG

(Magma) I:=[1, 0, 2, 2, 8, 22]; [n le 6 select I[n] else 4*Self(n1)5*Self(n2)+2*Self(n3): n in [1..40]]; // Vincenzo Librandi, Jun 28 2012


CROSSREFS

Cf. A005803, A060576.
Sequence in context: A208235 A151377 A151407 * A151384 A300460 A290613
Adjacent sequences: A130099 A130100 A130101 * A130103 A130104 A130105


KEYWORD

nonn,easy


AUTHOR

Paul Barry, May 07 2007


STATUS

approved



