A082134 Expansion of e.g.f. x*exp(3*x)*cosh(x).

0, 1, 6, 30, 144, 680, 3168, 14560, 66048, 296064, 1313280, 5772800, 25178112, 109078528, 469819392, 2013388800, 8590196736, 36507779072, 154620002304, 652837519360, 2748784312320, 11544883101696, 48378534690816

Offset: 0

Author

Paul Barry, Apr 06 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A082133. 3rd binomial transform of (0,1,0,3,0,5,0,7,...)

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

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
• Index entries for linear recurrences with constant coefficients, signature (12,-52,96,-64).

Crossrefs

Cf. A057711 (x*exp(x)*cosh(x)), A082133 (x*exp(2*x)*cosh(x)).

Cf. A082135 (x*exp(4*x)*cosh(x)), A082136 (x*exp(5*x)*cosh(x)).

Cf. A002697, A133224.

Programs

Magma

[n*2^(n-2)*(1+2^(n-1)): n in [0..25]]; // G. C. Greubel, Feb 05 2018

Maple

a:= n -> n*binomial(2^(n-1) +1, 2); seq(a(n), n=0..25); # G. C. Greubel, Apr 16 2020

Mathematica

Table[n(2^(n-1) +4^(n-1))/2, {n, 0, 22}] (* Michael De Vlieger, Nov 29 2015 *)
With[{nmax = 25}, CoefficientList[Series[x*Exp[3*x]*Cosh[x], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Feb 05 2018 *)

PARI

a(n)=n*(2^n--+4^n)/2 \\ Charles R Greathouse IV, Jan 14 2013

Sage

[n*binomial(2^(n-1)+1, 2) for n in (0..25)] # G. C. Greubel, Apr 16 2020

Formula

a(n) = n*(2^(n-1) + 4^(n-1))/2.

E.g.f.: x*exp(3*x)*cosh(x).

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

G.f.: x*(10*x^2-6*x+1) / ((2*x-1)^2*(4*x-1)^2). - Colin Barker, Dec 10 2012