A136303 Expansion of g.f. (1 +x^2)/((1-x)^2*(1 -3*x +2*x^2 -x^3)).

1, 5, 17, 48, 123, 300, 714, 1679, 3925, 9149, 21296, 49537, 115192, 267824, 622653, 1447533, 3365149, 7823068, 18186475, 42278476, 98285586, 228486323, 531166317, 1234811937, 2870589548, 6673311137, 15513566304, 36064666240, 83840177305

Offset: 0

Author

Richard Choulet, Mar 22 2008

Keywords

• nonn
• easy

Comments

Previous name: Transform of 0 by the reciprocal transformation to T_{1,1} (see link).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Richard Choulet, Curtz-like transformation.
• Index entries for linear recurrences with constant coefficients, signature (5,-9,8,-4,1).

Crossrefs

Cf. A095263, A097550, A135364, A136302, A136304, A136305, A137229, A137234, A137249.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x^2)/((1-x)^2*(1-3*x+2*x^2-x^3)) )); // G. C. Greubel, Apr 19 2021

Maple

A136303:= n-> -2*(n+2) + add( (5*binomial(n+k+2, 3*k+2) - 4*binomial(n +k+1, 3*k+2) + 2*binomial(n+k, 3*k+2)), k=0..n/2 );
seq(A136303(n), n=0..40); # G. C. Greubel, Apr 19 2021

Mathematica

LinearRecurrence[{5,-9,8,-4,1},{1,5,17,48,123},40] (* Harvey P. Dale, Apr 01 2018 *)

Sage

def A136303_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)/((1-x)^2*(1-3*x+2*x^2-x^3)) ).list()
A136303_list(40) # G. C. Greubel, Apr 19 2021

Formula

G.f.: f(z) = 1 +5*z + ... = (1+z^2)/((1-z)^2*(1-3*z+2*z^2-z^3)).

a(n+5) = 5*a(n+4) -9*a(n+3) +8*a(n+2) -4*a(n+1) +a(n) (n>=0). - Richard Choulet, Apr 07 2009

From G. C. Greubel, Apr 19 2021: (Start)

a(n) = -2*(n+2) + 5*A095263(n) - 4*A095263(n-1) + 2*A095263(n-2).

a(n) = -2*(n+2) + Sum_{k=0..floor(n/2)} (5*binomial(n+k+2, 3*k+2) - 4*binomial(n +k+1, 3*k+2) + 2*binomial(n+k, 3*k+2)). (End)