A128316 - OEIS

%I #17 Jun 23 2024 22:02:25

%S 1,1,1,3,-1,1,2,3,-2,1,4,-1,4,-3,1,4,3,-5,7,-4,1,6,-3,10,-13,11,-5,1,

%T 4,8,-14,23,-24,16,-6,1,7,-2,15,-33,46,-40,22,-7,1,7,4,-15,47,-79,86,

%U -62,29,-8,1,9,-6,30,-73,131,-166,148,-91,37,-9,1,7,12,-37,103,-204,297,-314,239,-128,46,-10,1

%N Triangle read by rows: A000012 * A128315 as infinite lower triangular matrices.

%C A128316 * [1,2,3...] = A000034: [1,2,1,2,...].

%H G. C. Greubel, <a href="/A128316/b128316.txt">Rows n = 1..100 of the triangle, flattened</a>

%F Sum_{k=1..n} T(n, k) = A000027(n) (row sums).

%F T(n, 1) = A059851(n).

%F From _G. C. Greubel_, Jun 23 2024: (Start)

%F T(n, k) = A010766(n,k) * AA130595(n-1, k-1) as infinite lower triangular matrices.

%F T(n, k) = Sum_{j=k..n} (-1)^(j+k) * floor(n/j) * binomial(j-1, k-1).

%F T(2*n-1, n) = (-1)^(n-1)*A026641(n).

%F T(2*n-2, n-1) = (-1)^n*A014300(n-1), for n >= 2.

%F Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A344817(n).

%F Sum_{k=1..n} k*T(n, k) = A032766(n-1).

%F Sum_{k=1..n} (k+1)*T(n, k) = A047215(n). (End)

%e First few rows of the triangle:

%e   1;

%e   1,  1;

%e   3, -1,   1;

%e   2,  3   -2,   1;

%e   4, -1,   4,  -3,   1;

%e   4,  3,  -5,   7,  -4,  1;

%e   6, -3,  10, -13,  11, -5,  1;

%e   4,  8, -14,  23, -24, 16, -6, 1;

%e   ...

%t T[n_, k_]:= Sum[(-1)^(j+k)*Floor[n/j]*Binomial[j-1,k-1], {j,k,n}];

%t Table[T[n,k], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Jun 23 2024 *)

%o (Magma)

%o A128316:= func< n,k | (&+[(-1)^(j+k)*Floor(n/j)*Binomial(j-1,k-1): j in [k..n]]) >;

%o [A128316(n,k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Jun 23 2024

%o (SageMath)

%o def A128316(n,k): return sum((-1)^(j+k)*int(n//j)*binomial(j-1,k-1) for j in range(k,n+1))

%o flatten([[A128316(n,k) for k in range(1,n+1)] for n in range(1,16)]) # _G. C. Greubel_, Jun 23 2024

%Y Cf. A000012, A000034, A014300, A026641, A059851, A128315.

%Y Sums include: A000027 (row), A032766, A047215, A344817 (alternating sign).

%K tabl,sign

%O 1,4

%A _Gary W. Adamson_, Feb 25 2007

%E a(28) = 1 inserted and more terms from _Georg Fischer_, Jun 06 2023