A343953 - OEIS

%I #6 May 05 2021 13:48:14

%S 0,0,1,0,1,4,0,1,2,9,0,1,2,5,16,0,1,2,3,8,25,0,1,2,3,6,13,36,0,1,2,3,

%T 4,9,18,49,0,1,2,3,4,7,12,25,64,0,1,2,3,4,5,10,17,32,81,0,1,2,3,4,5,8,

%U 13,22,41,100,0,1,2,3,4,5,6,11,16,27,50,121,0,1,2,3,4,5,6,9,14,21,34,61,144

%N Square array T(n,k), n>=1, k>=0, read by antidiagonals, where row n is the expansion of x*(1+x^n)/((1-x)^2*(1-x^n)).

%F G.f. of row n: x*(1+x^n)/((1-x)^2*(1-x^n)), some cross-referenced sequences omitting the factor x and the initial term 0.

%e Square array begins:

%e 0, 1, 4, 9,16,25,36,49,64,81,100,121,   ... (A000290)

%e 0, 1, 2, 5, 8,13,18,25,32,41, 50, 61,   ... (A000982)

%e 0, 1, 2, 3, 6, 9,12,17,22,27, 34, 41,   ... (A008810)

%e 0, 1, 2, 3, 4, 7,10,13,16,21, 26, 31,   ... (A008811)

%e 0, 1, 2, 3, 4, 5, 8,11,14,17, 20, 25,   ... (A008812)

%e 0, 1, 2, 3, 4, 5, 6, 9,12,15, 18, 21,   ... (A008813)

%e 0, 1, 2, 3, 4, 5, 6, 7,10,13, 16, 19,   ... (A008814)

%e 0, 1, 2, 3, 4, 5, 6, 7, 8,11, 14, 17,   ... (A008815)

%e 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15,   ... (A008816)

%e 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13,   ... (A008817)

%e 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14,... not in the OEIS

%e ...

%t nmax = 15;

%t ro[n_] := ro[n] = CoefficientList[x(1+x^n)/((1-x)^2 (1-x^n))+O[x]^nmax, x];

%t T[n_, k_] := ro[n][[k+1]];

%t Table[T[n-k, k], {n, 1, nmax}, {k, 0, n-1}]  // Flatten

%Y Cf. A000290, A000982, A008810, A008811, A008812, A008813, A008814, A008815, A008816, A008817.

%K nonn,tabl

%O 0,6

%A _Jean-François Alcover_, May 05 2021