A099577 Diagonal sums of triangle A099575.

1, 1, 2, 2, 6, 7, 13, 15, 38, 47, 85, 104, 245, 313, 558, 706, 1594, 2080, 3674, 4753, 10429, 13817, 24246, 31875, 68497, 91804, 160301, 213345, 451166, 610247, 1061413, 1426503, 2978230, 4058629, 7036859, 9533213, 19694622, 27007760

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. A099574, A099575.

Programs

Magma

[(&+[Binomial(n-k+Floor(k/2)+1, 1+Floor(k/2))*(1+Floor(k/2))/(n-k+1): k in [0..Floor(n/2)]]): n in [0..40]]; // G. C. Greubel, Jul 24 2022

Maple

A099577 := proc(n)
    local a,k ;
    a := 0 ;
    for k from 0 to floor(n/2) do
        a := a+add(binomial(n-k+j,j),j=0..floor(k/2)) ;
    end do:
    a ;
end proc:
seq(A099577(n),n=0..50); # R. J. Mathar, Nov 28 2014

Mathematica

Table[Sum[Binomial[n-k+Floor[k/2]+1, 1+Floor[k/2]]*(1+Floor[k/2])/(n-k+1), {k, 0, Floor[n/2]}], {n, 0, 40}] (* G. C. Greubel, Jul 24 2022 *)

SageMath

[sum( binomial(n-k+(k//2)+1, 1+(k//2))*(1+(k//2))/(n-k+1) for k in (0..(n//2)) ) for n in (0..40)] # G. C. Greubel, Jul 24 2022

Formula

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

a(n) = Sum_{k=0..floor(n/2)} binomial(n - k + floor(k/2) + 1, 1 + floor(k/2))*(1 + floor(k/2))/(n-k+1). - G. C. Greubel, Jul 24 2022