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A127159 Triangle T(n,k) with T(n,k) = A061554(n,k) + A107430(n,k).

%I A127159 #10 Sep 08 2022 08:45:29
%S A127159 2,2,2,3,2,3,4,4,4,4,7,5,8,5,7,11,11,10,10,11,11,21,16,21,12,21,16,21,
%T A127159 36,36,28,28,28,28,36,36,71,57,64,36,56,36,64,57,71,127,127,93,93,72,
%U A127159 72,93,93,127,127,253,211,220,130,165,90,165,130,220,211,253
%N A127159 Triangle T(n,k) with T(n,k) = A061554(n,k) + A107430(n,k).
%H A127159 G. C. Greubel, <a href="/A127159/b127159.txt">Rows n = 0..100 of triangle, flattened</a>
%F A127159 Sum_{k=0..n} T(n,k) = 2^(n+1).
%F A127159 T(n, k) = binomial(n, floor((n+1 - (-1)^(n-k)*(k+1))/2)) + binomial(n, floor(k/2)). - _G. C. Greubel_, Jan 31 2020
%e A127159 Triangle begins:
%e A127159    2;
%e A127159    2,  2;
%e A127159    3,  2,  3;
%e A127159    4,  4,  4,  4;
%e A127159    7,  5,  8,  5,  7;
%e A127159   11, 11, 10, 10, 11, 11;
%e A127159   21, 16, 21, 12, 21, 16, 21;
%e A127159   36, 36, 28, 28, 28, 28, 36, 36;
%e A127159   71, 57, 64, 36, 56, 36, 64, 57, 71; ...
%p A127159 seq(seq( binomial(n, floor((n+1-(-1)^(n-k)*(k+1))/2)) +binomial(n, floor(k/2)), k=0..n), n=0..12); # _G. C. Greubel_, Jan 31 2020
%t A127159 T[n_, k_]= Binomial[n, Floor[(n+1 -(-1)^(n-k)*(k+1))/2]] + Binomial[n, Floor[k/2]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 31 2020 *)
%o A127159 (PARI) T(n,k) = binomial(n, (n+1 -(-1)^(n-k)*(k+1))\2 ) + binomial(n, k\2); \\ _G. C. Greubel_, Jan 31 2020
%o A127159 (Magma) [Binomial(n, Floor((n+1 -(-1)^(n-k)*(k+1))/2)) + Binomial(n, Floor(k/2)): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 31 2020
%o A127159 (Sage) [[binomial(n, floor((n+1 -(-1)^(n-k)*(k+1))/2)) + binomial(n, floor(k/2)) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jan 31 2020
%o A127159 (GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n, Int((n+1 -(-1)^(n-k)*(k+1))/2)) + Binomial(n, Int(k/2)) ))); # _G. C. Greubel_, Jan 31 2020
%Y A127159 Cf. A061554, A107430.
%K A127159 nonn,tabl
%O A127159 0,1
%A A127159 _Philippe Deléham_, Mar 25 2007