OFFSET
0,2
COMMENTS
Also the cogrowth sequence of the 16-element group C4 X C4 = <S,T | S^4, T^4, [S,T]>. - Sean A. Irvine, Nov 09 2024
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..830
Index entries for linear recurrences with constant coefficients, signature (12,64).
FORMULA
a(n) = (1/2)*(-4)^n + (1/4)*16^n for n > 0.
Let b(n) = a(n) - 2^(4n)/4 then b(n+1) = 4*b(n) - Benoit Cloitre, May 27 2004
G.f.: (1 - 10*x - 16*x^2)/((1-16*x)*(1+4*x)). - Seiichi Manyama, Mar 15 2019
G.f.: ((cos(x) + cosh(x))/2)^2 = Sum_{n >= 0} a(n)*x(4*n)/(4*n)!. - Peter Bala, Jun 20 2022
MAPLE
a := n -> if n = 0 then 1 else 4^(n - 1)*(2*(-1)^n + 4^n) fi:
seq(a(n), n = 0..19); # Peter Luschny, Jul 02 2022
MATHEMATICA
Table[Sum[Binomial[4n, 4k], {k, 0, n}], {n, 0, 30}] (* or *) Join[{1}, LinearRecurrence[{12, 64}, {2, 72}, 30]] (* Harvey P. Dale, Apr 24 2011 *)
PROG
(PARI) a(n)=sum(k=0, n, binomial(4*n, 4*k))
(PARI) N=66; x='x+O('x^N); Vec((1-10*x-16*x^2)/((1-16*x)*(1+4*x))) \\ Seiichi Manyama, Mar 15 2019
CROSSREFS
KEYWORD
easy,nonn,changed
AUTHOR
Sebastian Gutierrez and Sarah Kolitz (skolitz(AT)mit.edu), May 15 2002
STATUS
approved