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%I #10 Mar 19 2023 02:46:31
%S 1,2,2,4,5,3,8,10,10,4,16,19,23,17,5,32,36,46,46,26,6,64,69,87,102,82,
%T 37,7,128,134,162,204,204,134,50,8,256,263,303,387,443,373,205,65,9,
%U 512,520,574,718,886,886,634,298,82,10
%N Triangle read by rows: matrix product A007318 * A051340.
%H G. C. Greubel, <a href="/A130265/b130265.txt">Rows n = 0..50 of the triangle, flattened</a>
%F Binomial transform of A051340.
%F From _G. C. Greubel_, Mar 18 2023: (Start)
%F T(n, k) = (k+1)*binomial(n,k) + Sum_{j=1..n-k} binomial(n, j+k).
%F T(n, k) = (k+1)*binomial(n,k) + binomial(n,k+1)*Hypergeometric2F1([1, k-n+1], [k+2], -1).
%F T(2*n, n) = (1/2)*T(2*n+1, n) = A258431(n+1).
%F Sum_{k=0..n} T(n, k) = A001787(n+1).
%F Sum_{k=0..n-1} T(n, k) = A058877(n+1), for n >= 1.
%F Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A084633(n). (End)
%e First few rows of the triangle are:
%e 1;
%e 2, 2;
%e 4, 5, 3;
%e 8, 10, 10, 4;
%e 16, 19, 23, 17, 5;
%e 32, 36, 46, 46, 26, 6;
%e 64, 69, 87, 102, 82, 37, 7;
%p A051340 := proc(n,k)
%p if k = n then
%p n+1 ;
%p elif k <= n then
%p 1;
%p else
%p 0;
%p end if;
%p end proc:
%p A130265 := proc(n,k)
%p add( binomial(n,j)*A051340(j,k),j=k..n) ;
%p end proc:
%p seq(seq(A130265(n,k),k=0..n),n=0..15) ; # _R. J. Mathar_, Aug 06 2016
%t T[n_, k_]:= (k+1)*Binomial[n,k] + Binomial[n,k+1]*Hypergeometric2F1[1, k-n+1, k+2, -1];
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 18 2023 *)
%o (Magma)
%o A130265:= func< n,k | k eq n select n+1 else (k+1)*Binomial(n,k) + (&+[Binomial(n, j+k): j in [1..n-k]]) >;
%o [A130265(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 18 2023
%o (SageMath)
%o def A130265(n,k): return (k+1)*binomial(n,k) + sum(binomial(n, j+k) for j in range(1,n-k+1))
%o flatten([[A130265(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Mar 18 2023
%Y Cf. A001787 (row sums), A007318, A051340, A058877, A084633, A258431.
%K nonn,tabl
%O 0,2
%A _Gary W. Adamson_, May 18 2007
%E Missing term inserted by _R. J. Mathar_, Aug 06 2016
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