A027959 a(n) = Sum_{k=m..n} T(k,n-k), where m = floor((n+1)/2); a(n) is the n-th diagonal-sum of left justified array T given by A027948.

1, 1, 2, 3, 5, 7, 12, 16, 27, 37, 59, 85, 129, 192, 285, 428, 634, 949, 1412, 2104, 3140, 4671, 6973, 10378, 15478, 23058, 34362, 51216, 76305, 113736, 169465, 252561, 376362, 560851, 835821, 1245503, 1856132, 2765976, 4121947

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,3,1,-3,-1,1).

Crossrefs

Cf. A027948.

Programs

GAP

a:=[1,1,2,3,5,7];; for n in [7..40] do a[n]:=3*a[n-2]+a[n-3] -3*a[n-4]-a[n-5]+a[n-6]; od; a; # G. C. Greubel, Sep 30 2019

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x*(1+x-x^2-x^3+x^4)/((1-x^2)*(1-2*x^2-x^3+x^4)) )); // G. C. Greubel, Sep 30 2019

Maple

seq(coeff(series(x*(1+x-x^2-x^3+x^4)/((1-x^2)*(1-2*x^2-x^3+x^4)), x, n+1), x, n), n = 1..40); # G. C. Greubel, Sep 30 2019

Mathematica

T[n_, k_]:= If[k==n, 1, Sum[Binomial[k+j, 2*j-1], {j, 0, n-k}]]; Table[Sum[T[k, n-k], {k, Floor[(n-1)/2], n}], {n,0,40}] (* G. C. Greubel, Sep 30 2019 *)

PARI

my(x='x+O('x^40)); Vec(x*(1+x-x^2-x^3+x^4)/((1-x^2)*(1-2*x^2-x^3+x^4))) \\ G. C. Greubel, Sep 30 2019

Sage

def A027959_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x-x^2-x^3+x^4)/((1-x^2)*(1-2*x^2-x^3+x^4)) ).list()
a=A027959_list(40); a[1:] # G. C. Greubel, Sep 30 2019

Formula

G.f.: x*(1+x-x^2-x^3+x^4)/((1-x)*(1+x)*(1-2*x^2-x^3+x^4)). - Colin Barker, Nov 25 2014