login
A070782
a(n) = Sum_{k=0..n} binomial(5*n,5*k).
16
1, 2, 254, 6008, 215766, 6643782, 215492564, 6863694378, 219993856006, 7035859329512, 225191238869774, 7205634556190798, 230585685502492596, 7378682274243863442, 236118494435702913134, 7555784484021765207768, 241785184867484394069286, 7737125013254912900576822
OFFSET
0,2
FORMULA
a(n) = (1/5)*32^n + (2/5)*(-11/2 + (5/2)*sqrt(5))^n + (2/5)*(-11/2 - (5/2)*sqrt(5))^n.
Let b(n) = a(n) - 2^(5n)/5; then b(n) + 11*b(n-1) - b(n-2) = 0. - Benoit Cloitre, May 27 2004
From Colin Barker, May 27 2019: (Start)
G.f.: (1 - 19*x - 141*x^2) / ((1 - 32*x)*(1 + 11*x - x^2)).
a(n) = 21*a(n-1) + 353*a(n-2) - 32*a(n-3) for n>2.
(End)
MATHEMATICA
LinearRecurrence[{21, 353, -32}, {1, 2, 254}, 20] (* Harvey P. Dale, Jun 18 2023 *)
PROG
(PARI) a(n)=sum(k=0, n, binomial(5*n, 5*k))
(PARI) Vec((1 - 19*x - 141*x^2) / ((1 - 32*x)*(1 + 11*x - x^2)) + O(x^20)) \\ Colin Barker, May 27 2019
CROSSREFS
Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), this sequence (b=5), A070967 (b=6), A094211 (b=7), A070832 (b=8), A094213 (b=9), A070833 (b=10).
Sequence in context: A304211 A224828 A370367 * A370965 A078167 A004875
KEYWORD
easy,nonn
AUTHOR
Sebastian Gutierrez and Sarah Kolitz (skolitz(AT)mit.edu), May 15 2002
STATUS
approved