A082308 Expansion of e.g.f. (1+x)*exp(4*x)*cosh(x).

1, 5, 25, 127, 657, 3449, 18281, 97395, 519841, 2773741, 14776377, 78538343, 416367665, 2201517153, 11610231433, 61078202971, 320570884929, 1678897264085, 8775159682649, 45780628812879, 238431945108433

Offset: 0

Author

Paul Barry, Apr 09 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A082307.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (16,-94,240,-225).

Crossrefs

Cf. A082309.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1+x)*Exp(4*x)*Cosh(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Sep 16 2018

Mathematica

With[{nmax = 50}, CoefficientList[Series[(1 + x)*Exp[4*x]*Cosh[x], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Sep 16 2018 *)

PARI

x='x+O('x^30); Vec(serlaplace((1+x)*exp(4*x)*cosh(x))) \\ G. C. Greubel, Sep 16 2018

Formula

a(n) = (A081105(n) + A006234(n))/2.

a(n) = ((n+3)*3^(n-1) + (n+5)*5^(n-1))/2.

G.f.: ((1-4*x)/(1-5*x)^2 + (1-2*x)/(1-3*x)^2)/2.

E.g.f.: (1+x)*exp(4*x)*cosh(x) = (1+x)*(exp(5*x) + exp(3*x))/2.