A084154 Binomial transform of sinh(x)*cosh(sqrt(2)*x).

0, 1, 2, 10, 32, 116, 392, 1352, 4608, 15760, 53792, 183712, 627200, 2141504, 7311488, 24963200, 85229568, 290992384, 993509888, 3392055808, 11581202432, 39540700160, 135000393728, 460920178688, 1573679923200, 5372879343616

Offset: 0

Author

Paul Barry, May 16 2003

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,0,-8,4).

Crossrefs

Cf. A084155.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1-2*x+2*x^2)/((1-2*x^2)*(1-4*x+2*x^2)))); // G. C. Greubel, Aug 16 2018

Mathematica

With[{nn=30},CoefficientList[Series[Exp[x]Sinh[x]Cosh[Sqrt[2]x],{x,0, nn}], x] Range[0,nn]!] (* or *) LinearRecurrence[{4,0,-8,4},{0,1,2, 10}, 30] (* Harvey P. Dale, Jun 19 2016 *)

PARI

x='x+O('x^30); concat([0], Vec(x*(1-2*x+2*x^2)/((1-2*x^2)*(1-4*x+2*x^2)))) \\ G. C. Greubel, Aug 16 2018

Formula

a(n) = 4*a(n-1) - 8*a(n-3) + 4*a(n-4), a(0)=0, a(1)=1, a(2)=2, a(3)=10.

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

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

E.g.f.: exp(x)*sinh(x)*cosh(sqrt(2)*x).