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A099577 Diagonal sums of triangle A099575.

%I A099577 #10 Jul 25 2022 01:13:12
%S A099577 1,1,2,2,6,7,13,15,38,47,85,104,245,313,558,706,1594,2080,3674,4753,
%T A099577 10429,13817,24246,31875,68497,91804,160301,213345,451166,610247,
%U A099577 1061413,1426503,2978230,4058629,7036859,9533213,19694622,27007760
%N A099577 Diagonal sums of triangle A099575.
%H A099577 G. C. Greubel, <a href="/A099577/b099577.txt">Table of n, a(n) for n = 0..1000</a>
%F A099577 a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..floor(k/2)} binomial(n-k+j, j).
%F A099577 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
%p A099577 A099577 := proc(n)
%p A099577     local a,k ;
%p A099577     a := 0 ;
%p A099577     for k from 0 to floor(n/2) do
%p A099577         a := a+add(binomial(n-k+j,j),j=0..floor(k/2)) ;
%p A099577     end do:
%p A099577     a ;
%p A099577 end proc:
%p A099577 seq(A099577(n),n=0..50); # _R. J. Mathar_, Nov 28 2014
%t A099577 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 *)
%o A099577 (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
%o A099577 (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
%Y A099577 Cf. A099574, A099575.
%K A099577 easy,nonn
%O A099577 0,3
%A A099577 _Paul Barry_, Oct 23 2004