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

Author

Paul Barry, Oct 23 2004

Keywords

• easy
• nonn

Links

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

Crossrefs

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

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^4)/((1-x^2-x^4)*(1-x-x^4)) )); // G. C. Greubel, Jul 25 2022

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 *)

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

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)