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%I #5 Feb 17 2021 20:23:57
%S 1,2,1,3,5,1,4,12,8,1,5,22,26,11,1,6,35,60,45,14,1,7,51,115,125,69,17,
%T 1,8,70,196,280,224,98,20,1,9,92,308,546,574,364,132,23,1,10,117,456,
%U 966,1260,1050,552,171,26,1
%N Triangle T(n, k) = binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1), read by rows.
%H G. C. Greubel, <a href="/A134081/b134081.txt">Rows n = 0..100 of the triangle, flattened</a>
%F Binomial transform of A112295(unsigned).
%F From _G. C. Greubel_, Feb 17 2021: (Start)
%F T(n, k) = binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1).
%F Sum_{k=0..n} T(n, k) = 2^n *n + 1 = A002064(n). (End)
%e First few rows of the triangle are:
%e 1;
%e 2, 1;
%e 3, 5, 1;
%e 4, 12, 8, 1;
%e 5, 22, 26, 11, 1;
%e 6, 35, 60, 45, 14, 1;
%e 7, 51, 115, 125, 69, 17, 1;
%e ...
%t T[n_, k_]:= Binomial[n, k]*((2*k+1)*(n-k) +k+1)/(k+1);
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 17 2021 *)
%o (Sage)
%o def A134081(n,k): return binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1)
%o flatten([[A134081(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 17 2021
%o (Magma)
%o A134081:= func< n,k | Binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1) >;
%o [A134081(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 17 2021
%Y Cf. A002064, A007318, A112295.
%Y Columns: A000027, A000326, A002413, A051740, A051879.
%K nonn,tabl
%O 0,2
%A _Gary W. Adamson_, Oct 07 2007
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