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A082134 Expansion of e.g.f. x*exp(3*x)*cosh(x). 8
0, 1, 6, 30, 144, 680, 3168, 14560, 66048, 296064, 1313280, 5772800, 25178112, 109078528, 469819392, 2013388800, 8590196736, 36507779072, 154620002304, 652837519360, 2748784312320, 11544883101696, 48378534690816 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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).

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

MATHEMATICA

Table[n (2^(n - 1) + 4^(n - 1))/2, {n, 0, 22}] (* Michael De Vlieger, Nov 29 2015 *)

With[{nmax = 50}, CoefficientList[Series[x*Exp[3*x]*Cosh[x], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Feb 05 2018 *)

PROG

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

(MAGMA) [n*(2^(n-1)+4^(n-1))/2: n in [0..30]]; // G. C. Greubel, Feb 05 2018

CROSSREFS

Cf. A057711, A082135.

Cf. A133224, A002697.

Sequence in context: A026749 A003279 A221397 * A030192 A026376 A026899

Adjacent sequences:  A082131 A082132 A082133 * A082135 A082136 A082137

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Apr 06 2003

STATUS

approved

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Last modified June 25 18:36 EDT 2019. Contains 324353 sequences. (Running on oeis4.)