A094213 a(n) = Sum_{k=0..n} binomial(9*n,9*k).

1, 2, 48622, 9373652, 9263421862, 3433541316152, 2140802758677844, 984101481334553024, 536617781178725122150, 265166261617029717011822, 138567978655457801631498052, 70126939586658252408697345838, 36144812798331420987905742371116

Offset: 0

Author

Benoit Cloitre, May 27 2004

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..369
• Index entries for linear recurrences with constant coefficients, signature (265,139823,-6826204,-6965249,512).

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), A094211 (b=7), A070832 (b=8), this sequence (b=9), A070833 (b=10).

Programs

Mathematica

Table[Sum[Binomial[9n,9k],{k,0,n}],{n,0,15}] (* Harvey P. Dale, Jul 14 2019 *)

PARI

a(n)=sum(k=0,n,binomial(9*n,9*k))

PARI

Vec((1 - 263*x - 91731*x^2 + 3035380*x^3 + 1547833*x^4) / ((1 + x)*(1 - 512*x)*(1 + 246*x - 13605*x^2 + x^3)) + O(x^15)) \\ Colin Barker, May 27 2019

Formula

Let b(n) = a(n)-2^(9*n)/9 then b(n)+246*b(n-1)-13605*b(n-2)+b(n-3)+(-1)^n*3078=0.

Conjectures from Colin Barker, May 27 2019: (Start)

G.f.: (1 - 263*x - 91731*x^2 + 3035380*x^3 + 1547833*x^4) / ((1 + x)*(1 - 512*x)*(1 + 246*x - 13605*x^2 + x^3)).

a(n) = 265*a(n-1) + 139823*a(n-2) - 6826204*a(n-3) - 6965249*a(n-4) + 512*a(n-5) for n>4. (End)

a(n) ~ (1/9)*exp(n*9*log(2)) (conjecture). - Bill McEachen, Aug 11 2025