A070775 a(n) = Sum_{k=0..n} binomial(4*n,4*k).

1, 2, 72, 992, 16512, 261632, 4196352, 67100672, 1073774592, 17179738112, 274878431232, 4398044413952, 70368752566272, 1125899873288192, 18014398643699712, 288230375614840832, 4611686020574871552, 73786976286248271872, 1180591620751771041792, 18889465931341141901312

Offset: 0

Author

Sebastian Gutierrez and Sarah Kolitz (skolitz(AT)mit.edu), May 15 2002

Keywords

• easy
• nonn

Comments

Also the cogrowth sequence of the 16-element group C4 X C4 = . - 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).

Crossrefs

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), this sequence (b=4), A070782 (b=5), A070967 (b=6), A094211 (b=7), A070832 (b=8), A094213 (b=9), A070833 (b=10).

Programs

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 *)

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

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