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)).
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, 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, 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
Offset: 0
Examples
Square array begins: 0, 1, 4, 9,16,25,36,49,64,81,100,121, ... (A000290) 0, 1, 2, 5, 8,13,18,25,32,41, 50, 61, ... (A000982) 0, 1, 2, 3, 6, 9,12,17,22,27, 34, 41, ... (A008810) 0, 1, 2, 3, 4, 7,10,13,16,21, 26, 31, ... (A008811) 0, 1, 2, 3, 4, 5, 8,11,14,17, 20, 25, ... (A008812) 0, 1, 2, 3, 4, 5, 6, 9,12,15, 18, 21, ... (A008813) 0, 1, 2, 3, 4, 5, 6, 7,10,13, 16, 19, ... (A008814) 0, 1, 2, 3, 4, 5, 6, 7, 8,11, 14, 17, ... (A008815) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15, ... (A008816) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, ... (A008817) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14,... not in the OEIS ...
Crossrefs
Programs
-
Mathematica
nmax = 15; ro[n_] := ro[n] = CoefficientList[x(1+x^n)/((1-x)^2 (1-x^n))+O[x]^nmax, x]; T[n_, k_] := ro[n][[k+1]]; Table[T[n-k, k], {n, 1, nmax}, {k, 0, n-1}] // Flatten
Formula
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.