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A130265 Triangle read by rows: matrix product A007318 * A051340.

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