A137360 a(n) = Sum_{k <= n/2 } k*binomial(n-2k, 3k+1).

0, 0, 0, 0, 0, 0, 1, 5, 15, 35, 70, 128, 226, 402, 735, 1375, 2588, 4830, 8882, 16108, 28943, 51785, 92573, 165525, 295869, 528069, 940259, 1669725, 2957941, 5229953, 9233748, 16284106, 28688451, 50490125, 88765885, 155891305, 273495479, 479360847, 839451764

Offset

0,8

References

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Links

Table of n, a(n) for n=0..38.

Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 8, -7, 6, -2, 0, -1).

Formula

G.f.: x^6*(1-x)/(x^5+x^3-3*x^2+3*x-1)^2. - Alois P. Heinz, Oct 23 2008

Maple

a:= n-> (Matrix([[35, 15, 5, 1, 0$6]]). Matrix (10, (i, j)-> if i=j-1 then 1 elif j=1 then [6, -15, 20, -15, 8, -7, 6, -2, 0, -1][i] else 0 fi)^n)[1, 10]: seq (a(n), n=0..50);  # Alois P. Heinz, Oct 23 2008

Mathematica

Table[Sum[k Binomial[n-2k, 3k+1], {k, n/2}], {n, 0, 40}] (* or *) LinearRecurrence[ {6, -15, 20, -15, 8, -7, 6, -2, 0, -1}, {0, 0, 0, 0, 0, 0, 1, 5, 15, 35}, 40] (* Harvey P. Dale, May 31 2017 *)

Prog

(PARI) Vec(x^6*(1-x)/(x^5+x^3-3*x^2+3*x-1)^2+O(x^99)) \\ Charles R Greathouse IV, May 26 2026

Crossrefs

Cf. A137356-A137361, A136444.

Sequence in context: A360047 A373964 A275935 * A195760 A195761 A100355

Adjacent sequences: A137357 A137358 A137359 * A137361 A137362 A137363

Keyword

nonn,easy

Author

Don Knuth, Apr 11 2008

Status

approved