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

0,6

Links

Table of n, a(n) for n=0..90.

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.

Example

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
...

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

Crossrefs

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

Sequence in context: A179296 A196817 A229987 * A307769 A096793 A155998

Adjacent sequences: A343950 A343951 A343952 * A343954 A343955 A343956

Keyword

nonn,tabl

Author

Jean-François Alcover, May 05 2021

Status

approved