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Triangle T(n, k) = binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1), read by rows.
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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