A070967 a(n) = Sum_{k=0..n} binomial(6*n,6*k).

1, 2, 926, 37130, 2973350, 174174002, 11582386286, 729520967450, 47006639297270, 2999857885752002, 192222214478506046, 12295976362284182570, 787111112023373201990, 50370558298891875954002, 3223838658635388303336206, 206322355109994528871954490

Offset: 0

Author

Sebastian Gutierrez and Sarah Kolitz (skolitz(AT)mit.edu), May 16 2002

Keywords

• easy
• nonn

References

• Matthijs Coster, Supercongruences, Thesis, Jun 08, 1988.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..554
• Index entries for linear recurrences with constant coefficients, signature (38,1691,-1728).

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), this sequence (b=6), A094211 (b=7), A070832 (b=8), A094213 (b=9), A070833 (b=10).

Programs

Mathematica

Table[Sum[Binomial[6n,6k],{k,0,n}],{n,0,20}] (* or *) LinearRecurrence[ {38,1691,-1728},{1,2,926,37130},30] (* Harvey P. Dale, Jun 19 2021 *)

PARI

a(n)=sum(k=0,n,binomial(6*n,6*k))

PARI

a(n)=if(n<0,0,(2*(-27)^n+2+64^n+0^n)/6)

PARI

a(n)=if(n<0,0,polsym(x*(x-64)*(x+27)^2*(x-1)^2,n)[n+1]/6)

Formula

G.f.: (1-36x-841x^2+288x^3)/((1-x)*(1+27x)*(1-64x)).

a(n) = ((-27)^n + 1)/3 + (64^n + 0^n)/6.

Let b(n) = a(n)-2^(6n)/6 then b(n)+26*b(n-1)-27*b(n-2) = 0. - Benoit Cloitre, May 27 2004