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A134081 Triangle T(n, k) = binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1), read by rows.

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