cp's OEIS Frontend

A345033 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-k)^(floor(n/j) - 1).

%I A345033 #13 Jun 06 2021 08:59:32
%S A345033 1,1,1,1,0,2,1,-1,3,2,1,-2,6,0,3,1,-3,11,-8,3,3,1,-4,18,-28,17,2,4,1,
%T A345033 -5,27,-66,81,-27,5,4,1,-6,38,-128,255,-234,70,0,5,1,-7,51,-220,623,
%U A345033 -1009,739,-136,5,5,1,-8,66,-348,1293,-3102,4112,-2216,255,4,6,1,-9,83,-518,2397,-7743,15649,-16452,6545,-491,7,6
%N A345033 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-k)^(floor(n/j) - 1).
%F A345033 G.f. of column k: (1/(1 - x)) * Sum_{j>=1} x^j * (1 - x^j)/(1 + k*x^j).
%e A345033 Square array begins:
%e A345033   1, 1,   1,    1,     1,     1,     1, ...
%e A345033   1, 0,  -1,   -2,    -3,    -4,    -5, ...
%e A345033   2, 3,   6,   11,    18,    27,    38, ...
%e A345033   2, 0,  -8,  -28,   -66,  -128,  -220, ...
%e A345033   3, 3,  17,   81,   255,   623,  1293, ...
%e A345033   3, 2, -27, -234, -1009, -3102, -7743, ...
%t A345033 T[n_, 0] := Floor[(n + 1)/2]; T[n_, k_] := Sum[(-k)^(Floor[n/j] - 1), {j, 1, n}]; Table[T[k, n - k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Jun 06 2021 *)
%o A345033 (PARI) T(n, k) = sum(j=1, n, (-k)^(n\j-1));
%Y A345033 Columns k=0..3 give A110654, A271860, A345034, A345035.
%Y A345033 T(n,n) gives A345036.
%Y A345033 Cf. A344824, A345032.
%K A345033 sign,tabl
%O A345033 1,6
%A A345033 _Seiichi Manyama_, Jun 06 2021