OFFSET
0,2
COMMENTS
Binomial transform is A014916. In general, ((1+x)/(1-r*x))^2 expands to a(n) = ((r+1)*r^n*((r+1)*n + r-1) + 0^n)/r^2, which is also a(n) = Sum_{k=0..n} C(n,k)*Sum_{j=0..k} (j+1)*(r+1)^j. This is the self-convolution of the coordination sequence for the infinite tree with valency r.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-9).
FORMULA
G.f.: (1+x)^2/(1-3*x)^2. [Corrected by Georg Fischer, May 24 2019]
a(n) = (8*3^n*(2*n+1) + 0^n)/9 = (4*3^n*(4*n+2) + 0^n)/9;
a(n) = Sum_{k=0..n} C(n, k)*Sum_{j=0..k} (j+1)*4^j.
a(n) = 8*A081038(n-1), n>0. - R. J. Mathar, Nov 28 2014
E.g.f.: (1 + 8*exp(3*x)*(1 + 6*x))/9. - Stefano Spezia, Jan 31 2025
MATHEMATICA
CoefficientList[Series[(1+x)^2/(1-3x)^2, {x, 0, 30}], x] (* Georg Fischer, May 24 2019 *)
LinearRecurrence[{6, -9}, {1, 8, 40}, 30] (* G. C. Greubel, May 24 2019 *)
PROG
(PARI) my(x='x+O('x^30)); Vec(((1+x)/(1-3*x))^2) \\ G. C. Greubel, May 24 2019
(Magma) I:=[8, 40]; [1] cat [n le 2 select I[n] else 6*Self(n-1) - 9*Self(n-2): n in [1..30]]; // G. C. Greubel, May 24 2019
(Sage) (((1+x)/(1-3*x))^2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019
(GAP) a:=[1, 8, 40];; for n in [4..30] do a[n]:=6*a[n-1]-9*a[n-2]; od; a; # G. C. Greubel, May 24 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Oct 14 2005
STATUS
approved