A293169 a(n) = Sum_{k=0..n} binomial(k, 6*(n-k)).

1, 1, 1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 463, 925, 1718, 3017, 5097, 8464, 14197, 24753, 45697, 89150, 180254, 368734, 748924, 1493990, 2914906, 5565127, 10434412, 19322901, 35583926, 65615746, 121847272, 228638698, 433747259, 830227401, 1597653852, 3078928619, 5922703731, 11347651254

Offset

0,8

Links

Colin Barker, Table of n, a(n) for n = 0..1000

V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1,1).

Formula

From Colin Barker, Oct 17 2017: (Start)

G.f.: (1 - x)^5 / (1 - 6*x + 15*x^2 - 20*x^3 + 15*x^4 - 6*x^5 + x^6 - x^7).

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) + a(n-7) for n>6.

(End)

Maple

f:=n-> add( binomial(k, 6*(n-k)), k=0..n);
[seq(f(n), n=0..30)];

Mathematica

Table[Sum[Binomial[k, 6(n-k)], {k, 0, n}], {n, 0, 40}] (* or *)  LinearRecurrence[{6, -15, 20, -15, 6, -1, 1}, {1, 1, 1, 1, 1, 1, 1}, 50] (* Harvey P. Dale, Apr 10 2022 *)

Prog

(PARI) Vec((1 - x)^5 / (1 - 6*x + 15*x^2 - 20*x^3 + 15*x^4 - 6*x^5 + x^6 - x^7) + O(x^30)) \\ Colin Barker, Oct 18 2017

Crossrefs

Cf. A005676, A099132.

Sequence in context: A261559 A061230 A241627 * A306847 A364523 A107025

Adjacent sequences: A293166 A293167 A293168 * A293170 A293171 A293172

Keyword

nonn,easy

Author

N. J. A. Sloane, Oct 17 2017

Status

approved