|
| |
|
|
A113070
|
|
Expansion of ((1+x)/(1-2x))^2.
|
|
2
| |
|
|
1, 6, 21, 60, 156, 384, 912, 2112, 4800, 10752, 23808, 52224, 113664, 245760, 528384, 1130496, 2408448, 5111808, 10813440, 22806528, 47972352, 100663296, 210763776, 440401920, 918552576, 1912602624, 3976200192, 8254390272, 17112760320
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| Binomial transform is A014915. 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
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
|
|
|
FORMULA
| G.f.: (1+x)^2/(1-2x)^2; a(n)=3*2^n(3n+1)/4+0^n/4; a(n)=sum{k=0..n, A003945(k)A003945(n-k)}; a(n)=sum{k=0..n, C(n, k)*sum{j=0..k, (j+1)*3^j}}.
a(0)=1, a(1)=6, a(2)=21, For n>2, a(n)=4*a(n-1)-4*a(n-2) [From Harvey P. Dale, May 20 2011]
|
|
|
MATHEMATICA
| Join[{1}, LinearRecurrence[{4, -4}, {6, 21}, 30]] (* or *) CoefficientList[ Series[((1+x)/(1-2x))^2, {x, 0, 30}], x] (* From Harvey P. Dale, May 20 2011 *)
|
|
|
PROG
| (MAGMA) [3*2^n*(3*n+1)/4+0^n/4: n in [0..30]]; // Vincenzo Librandi, May 21 2011
|
|
|
CROSSREFS
| Cf. A113071.
Sequence in context: A047520 A143115 A066524 * A009147 A012593 A048476
Adjacent sequences: A113067 A113068 A113069 * A113071 A113072 A113073
|
|
|
KEYWORD
| easy,nonn
|
|
|
AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Oct 14 2005
|
| |
|
|
