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%I #48 Sep 08 2022 08:45:09
%S 0,1,6,30,144,680,3168,14560,66048,296064,1313280,5772800,25178112,
%T 109078528,469819392,2013388800,8590196736,36507779072,154620002304,
%U 652837519360,2748784312320,11544883101696,48378534690816
%N Expansion of e.g.f. x*exp(3*x)*cosh(x).
%C Binomial transform of A082133. 3rd binomial transform of (0,1,0,3,0,5,0,7,...)
%C Let P(A) be the power set of an n-element set A and B be the Cartesian product of P(A) with itself. Then remove (y,x) from B when (x,y) is in B and x <> y and call this R35. Then a(n) = the sum of the size of the intersection of x and y for every (x,y) of R35. - _Ross La Haye_, Dec 30 2007; edited Jan 05 2013
%C A133224 is the analogous sequence if "Intersection" is replaced by "Union" and A002697 is the analogous sequence if "Intersection" is replaced by "Symmetric difference". Here, X Intersection Y = Y Intersection X is considered as the same set [Relation (37): T_Q(n) in document of Ross La Haye in reference]. If we want to consider that X Intersection Y and Y Intersection X are two distinct formula for describing the same set, see A002697. - _Bernard Schott_, Jan 19 2013
%H G. C. Greubel, <a href="/A082134/b082134.txt">Table of n, a(n) for n = 0..1000</a>
%H Ross La Haye, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/LaHaye/lahaye5.html">Binary Relations on the Power Set of an n-Element Set</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (12,-52,96,-64).
%F a(n) = n*(2^(n-1) + 4^(n-1))/2.
%F E.g.f.: x*exp(3*x)*cosh(x).
%F Conjecture: (n+28)*a(n) + (n-282)*a(n-1) + 2*(-17*n+423)*a(n-2) + 8*(7*n-94)*a(n-3) = 0. - _R. J. Mathar_, Nov 29 2012
%F G.f.: x*(10*x^2-6*x+1) / ((2*x-1)^2*(4*x-1)^2). - _Colin Barker_, Dec 10 2012
%p a:= n -> n*binomial(2^(n-1) +1, 2); seq(a(n), n=0..25); # _G. C. Greubel_, Apr 16 2020
%t Table[n(2^(n-1) +4^(n-1))/2, {n, 0, 22}] (* _Michael De Vlieger_, Nov 29 2015 *)
%t With[{nmax = 25}, CoefficientList[Series[x*Exp[3*x]*Cosh[x], {x, 0, nmax}], x]*Range[0, nmax]!] (* _G. C. Greubel_, Feb 05 2018 *)
%o (PARI) a(n)=n*(2^n--+4^n)/2 \\ _Charles R Greathouse IV_, Jan 14 2013
%o (Magma) [n*2^(n-2)*(1+2^(n-1)): n in [0..25]]; // _G. C. Greubel_, Feb 05 2018
%o (Sage) [n*binomial(2^(n-1)+1, 2) for n in (0..25)] # _G. C. Greubel_, Apr 16 2020
%Y Cf. A057711 (x*exp(x)*cosh(x)), A082133 (x*exp(2*x)*cosh(x)).
%Y Cf. A082135 (x*exp(4*x)*cosh(x)), A082136 (x*exp(5*x)*cosh(x)).
%Y Cf. A002697, A133224.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Apr 06 2003
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