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A099575 Number triangle T(n,k) = binomial(n + floor(k/2) + 1, n + 1), 0 <= k <= n.

%I A099575 #18 Jul 25 2022 04:02:54
%S A099575 1,1,1,1,1,4,1,1,5,5,1,1,6,6,21,1,1,7,7,28,28,1,1,8,8,36,36,120,1,1,9,
%T A099575 9,45,45,165,165,1,1,10,10,55,55,220,220,715,1,1,11,11,66,66,286,286,
%U A099575 1001,1001,1,1,12,12,78,78,364,364,1365,1365,4368,1,1,13,13,91,91,455,455,1820,1820,6188,6188
%N A099575 Number triangle T(n,k) = binomial(n + floor(k/2) + 1, n + 1), 0 <= k <= n.
%C A099575 Original name was: "Number triangle T(n,k) = if(k<=n, Sum_{j=0..floor(k/2)} binomial(n+j,j), 0)."
%H A099575 Robert Israel, <a href="/A099575/b099575.txt">Table of n, a(n) for n = 0..10010</a> (Rows 0..140, flattened)
%F A099575 T(n, k) = binomial(n + floor(k/2) + 1, n + 1).
%F A099575 T(n, n) = A099578(n).
%F A099575 Sum_{k=0..n} T(n, k) = A099576(n).
%F A099575 Sum_{k=0..floor(n/2)} T(n-k, k) = A099577(n).
%e A099575 Rows start:
%e A099575   1;
%e A099575   1, 1;
%e A099575   1, 1,  4;
%e A099575   1, 1,  5,  5;
%e A099575   1, 1,  6,  6, 21;
%e A099575   1, 1,  7,  7, 28, 28;
%e A099575   1, 1,  8,  8, 36, 36, 120;
%e A099575   1, 1,  9,  9, 45, 45, 165, 165;
%e A099575   1, 1, 10, 10, 55, 55, 220, 220, 715;
%p A099575 for n from 0 to 20 do seq(binomial(n+floor(k/2)+1,n+1),k=0..n) od; # _Robert Israel_, May 08 2018
%t A099575 Table[Binomial[n+Floor[k/2]+1, n+1], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 24 2022 *)
%o A099575 (Magma) [Binomial(n+1+Floor(k/2), n+1): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Jul 24 2022
%o A099575 (SageMath) flatten([[binomial(n+(k//2)+1, n+1) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Jul 24 2022
%Y A099575 Cf. A099573, A099576 (row sums), A099577 (diagonal sums), A099578 (main diagonal).
%K A099575 easy,nonn,tabl
%O A099575 0,6
%A A099575 _Paul Barry_, Oct 23 2004
%E A099575 Definition simplified by _Robert Israel_, May 08 2018