cp's OEIS Frontend

A350161 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-1)^(j+1) * floor(n/(2*j-1))^k.

%I A350161 #17 Dec 18 2021 11:17:17
%S A350161 1,1,2,1,4,2,1,8,8,3,1,16,26,15,5,1,32,80,63,25,5,1,64,242,255,125,33,
%T A350161 5,1,128,728,1023,625,209,45,6,1,256,2186,4095,3125,1281,335,60,7,1,
%U A350161 512,6560,16383,15625,7745,2385,504,73,9,1,1024,19682,65535,78125,46593,16775,4080,703,95,9
%N A350161 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-1)^(j+1) * floor(n/(2*j-1))^k.
%F A350161 G.f. of column k: (1/(1 - x)) * Sum_{j>=1} (j^k - (j - 1)^k) * x^j/(1 + x^(2*j)).
%F A350161 T(n,k) = Sum_{j=1..n} Sum_{d|j} A101455(j/d) * (d^k - (d - 1)^k).
%e A350161 Square array begins:
%e A350161   1,  1,   1,    1,     1,      1,      1, ...
%e A350161   2,  4,   8,   16,    32,     64,    128, ...
%e A350161   2,  8,  26,   80,   242,    728,   2186, ...
%e A350161   3, 15,  63,  255,  1023,   4095,  16383, ...
%e A350161   5, 25, 125,  625,  3125,  15625,  78125, ...
%e A350161   5, 33, 209, 1281,  7745,  46593, 279809, ...
%e A350161   5, 45, 335, 2385, 16775, 117585, 823415, ...
%t A350161 T[n_, k_] := Sum[(-1)^(j + 1) * Floor[n/(2*j - 1)]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Dec 18 2021 *)
%o A350161 (PARI) T(n, k) = sum(j=1, n, (-1)^(j+1)*(n\(2*j-1))^k);
%o A350161 (PARI) T(n, k) = sum(j=1, n, sumdiv(j, d, kronecker(-4, j/d)*(d^k-(d-1)^k)));
%Y A350161 Columns k=1..3 give A014200, A350162, A350163.
%Y A350161 T(n,n) gives A350164.
%Y A350161 Cf. A101455, A344726, A350122.
%K A350161 nonn,tabl
%O A350161 1,3
%A A350161 _Seiichi Manyama_, Dec 18 2021