A094211 a(n) = Sum_{k=0..n} binomial(7*n,7*k).

1, 2, 3434, 232562, 42484682, 4653367842, 644032289258, 79450506979090, 10353832741654602, 1313930226050847938, 168883831255263816554, 21573903987107973878962, 2764126124873404346104778, 353643666623193292098680930, 45276535087893983968685884906

Offset: 0

Author

Benoit Cloitre, May 27 2004

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..474
• Index entries for linear recurrences with constant coefficients, signature (71,7585,-36991,-128).

Crossrefs

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), A070782 (b=5), A070967 (b=6), this sequence (b=7), A070832 (b=8), A094213 (b=9), A070833 (b=10).

Programs

Maple

A094211:=n->add(binomial(7*n,7*k), k=0..n): seq(A094211(n), n=0..20); # Wesley Ivan Hurt, Feb 16 2017

Mathematica

Table[Sum[Binomial[7n,7k],{k,0,n}],{n,0,20}] (* or *) LinearRecurrence[ {71,7585,-36991,-128},{1,2,3434,232562},20] (* Harvey P. Dale, May 06 2012 *)

PARI

a(n)=sum(k=0,n,binomial(7*n,7*k))

Formula

Let b(n) = a(n) - 2^(7n)/7; then b(n) + 57*b(n-1) - 289*b(n-2) - b(n-3) = 0.

a(n) = 71*a(n-1) + 7585*a(n-2) - 36991*a(n-3) - 128*a(n-4); a(0)=1, a(1)=2, a(2)=3434, a(3)=232562. - Harvey P. Dale, May 06 2012

G.f.: (1 - 69*x - 4293*x^2 + 10569*x^3) / ((1 - 128*x)*(1 + 57*x - 289*x^2 - x^3)). - Colin Barker, May 27 2019

a(n) = (1 + 2*(s*(3 - 4*s^2))^(7*n) + 2*(-1)^n*((1 - 2*s^2)^(7*n) + s^(7*n))) * 2^(7*n)/7, where s = sin(Pi/14). - Vaclav Kotesovec, Apr 17 2023