A099574 Diagonal sums of triangle A099573.

1, 1, 2, 2, 4, 5, 9, 11, 18, 23, 37, 48, 74, 97, 147, 195, 290, 387, 568, 763, 1108, 1495, 2152, 2915, 4167, 5662, 8047, 10962, 15506, 21168, 29825, 40787, 57280, 78448, 109870, 150657, 210521, 288969, 403020, 553677, 770963, 1059932, 1473898

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..floor(k/2)} binomial(n-k-j, j).

G.f.: (1-x)*(1+x)*(1+x^2) / ( (1-x-x^4)*(1-x^2-x^4) ). - R. J. Mathar, Nov 11 2014

From G. C. Greubel, Jul 25 2022: (Start)

a(n) = A003269(n+5) - A079977(n+3) - A079977(n+2).

a(n) = A003269(n+5) - A103609(n+5). (End)

Mathematica

a[n_]:= a[n]= Sum[Binomial[n-k-j, j], {k, 0, Floor[n/2]}, {j, 0, Floor[k/2]}];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Jul 25 2022 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^4)/((1-x^2-x^4)*(1-x-x^4)) )); // G. C. Greubel, Jul 25 2022
(SageMath)
@CachedFunction
def A099574(n): return sum(sum(binomial(n-k-j, j) for j in (0..(k//2))) for k in (0..(n//2)))
[A099574(n) for n in (0..40)] # G. C. Greubel, Jul 25 2022

Crossrefs

Cf. A003269, A079977, A099573, A099577, A103609.

Sequence in context: A326632 A240206 A229816 * A144118 A187069 A038000

Adjacent sequences: A099571 A099572 A099573 * A099575 A099576 A099577

Keyword

easy,nonn

Author

Paul Barry, Oct 23 2004

Status

approved