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%I #6 Sep 08 2022 08:45:52
%S 1,1,1,1,6,1,1,6,6,1,1,12,30,12,1,1,10,30,30,10,1,1,18,60,140,60,18,1,
%T 1,14,105,140,140,105,14,1,1,24,84,280,630,280,84,24,1,1,18,144,504,
%U 630,630,504,144,18,1,1,30,225,840,1260,2772,1260,840,225,30,1
%N Triangle, read by rows: T(n, k) = binomial(n, k)*(1 + 2*(n+1) - (k+1)*floor((n+1)/(k+1)) - (n-k+1)* floor((n+1)/(n-k+1))).
%C Row sums are: {1, 2, 8, 14, 56, 82, 298, 520, 1408, 2594, 7484, ...}.
%H G. C. Greubel, <a href="/A176348/b176348.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n, k) = binomial(n, k)*(1 +2*(n+1) -(k+1)*floor((n+1)/(k+1)) -(n-k+1)* floor((n+1)/(n-k+1))).
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 6, 1;
%e 1, 6, 6, 1;
%e 1, 12, 30, 12, 1;
%e 1, 10, 30, 30, 10, 1;
%e 1, 18, 60, 140, 60, 18, 1;
%e 1, 14, 105, 140, 140, 105, 14, 1;
%e 1, 24, 84, 280, 630, 280, 84, 24, 1;
%e 1, 18, 144, 504, 630, 630, 504, 144, 18, 1;
%e 1, 30, 225, 840, 1260, 2772, 1260, 840, 225, 30, 1;
%p T:=binomial(n, k)*(2*n+3 -(k+1)*floor((n+1)/(k+1)) -(n-k+1)* floor((n+1)/(n-k+1))); seq(seq(T(n,k), k=0..n), n=0..12); # _G. C. Greubel_, Nov 23 2019
%t T[n_, k_]:= T[n, k]= Binomial[n,k]*(2*n+3 -(k+1)*Floor[(n+1)/(k+1)] -(n - k+1)*Floor[(n+1)/(n-k+1)]); Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
%o (PARI) T(n,k) = binomial(n, k)*(2*n+3 -(k+1)*((n+1)\(k+1)) -(n-k+1)* ((n+1)\(n-k+1))); \\ _G. C. Greubel_, Nov 23 2019
%o (Magma) [Binomial(n, k)*(2*n+3 -(k+1)*Floor((n+1)/(k+1)) -(n-k+1)* Floor((n+1)/(n-k+1))): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 23 2019
%o (Sage) [[binomial(n, k)*(2*n+3 -(k+1)*floor((n+1)/(k+1)) -(n-k+1)* floor((n+1)/(n-k+1))) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 23 2019
%o (GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n, k)*(2*n+3 -(k+1)*Int((n+1)/(k+1)) -(n-k+1)*Int((n+1)/(n-k+1))) ))); # _G. C. Greubel_, Nov 23 2019
%Y Cf. A007318, A176298.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Apr 15 2010
%E Edited by _G. C. Greubel_, Nov 23 2019
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