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A343953 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)).

%I A343953 #6 May 05 2021 13:48:14
%S A343953 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 A343953 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 A343953 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 A343953 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 A343953 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 A343953 Square array begins:
%e A343953 0, 1, 4, 9,16,25,36,49,64,81,100,121,   ... (A000290)
%e A343953 0, 1, 2, 5, 8,13,18,25,32,41, 50, 61,   ... (A000982)
%e A343953 0, 1, 2, 3, 6, 9,12,17,22,27, 34, 41,   ... (A008810)
%e A343953 0, 1, 2, 3, 4, 7,10,13,16,21, 26, 31,   ... (A008811)
%e A343953 0, 1, 2, 3, 4, 5, 8,11,14,17, 20, 25,   ... (A008812)
%e A343953 0, 1, 2, 3, 4, 5, 6, 9,12,15, 18, 21,   ... (A008813)
%e A343953 0, 1, 2, 3, 4, 5, 6, 7,10,13, 16, 19,   ... (A008814)
%e A343953 0, 1, 2, 3, 4, 5, 6, 7, 8,11, 14, 17,   ... (A008815)
%e A343953 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15,   ... (A008816)
%e A343953 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13,   ... (A008817)
%e A343953 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14,... not in the OEIS
%e A343953 ...
%t A343953 nmax = 15;
%t A343953 ro[n_] := ro[n] = CoefficientList[x(1+x^n)/((1-x)^2 (1-x^n))+O[x]^nmax, x];
%t A343953 T[n_, k_] := ro[n][[k+1]];
%t A343953 Table[T[n-k, k], {n, 1, nmax}, {k, 0, n-1}]  // Flatten
%Y A343953 Cf. A000290, A000982, A008810, A008811, A008812, A008813, A008814, A008815, A008816, A008817.
%K A343953 nonn,tabl
%O A343953 0,6
%A A343953 _Jean-François Alcover_, May 05 2021