A321877 - OEIS

%I #10 Nov 23 2018 04:06:23

%S 1,1,1,1,1,2,1,1,3,4,1,1,5,7,6,1,1,9,15,14,10,1,1,17,37,41,28,17,1,1,

%T 33,99,137,107,58,25,1,1,65,277,491,487,286,106,38,1,1,129,795,1829,

%U 2429,1749,700,201,59,1,1,257,2317,6971,12763,12056,5901,1735,372,86

%N Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)^sigma_k(j).

%H Seiichi Manyama, <a href="/A321877/b321877.txt">Antidiagonals n = 0..139, flattened</a>

%F G.f. of column k: Product_{i>=1, j>=1} (1 + x^(i*j))^(j^k).

%F G.f. of column k: exp(Sum_{j>=1} sigma_(k+1)(j)*x^j/(j*(1 - x^(2*j)))).

%e Square array begins:

%e    1,   1,    1,    1,     1,      1,  ...

%e    1,   1,    1,    1,     1,      1,  ...

%e    2,   3,    5,    9,    17,     33,  ...

%e    4,   7,   15,   37,    99,    277,  ...

%e    6,  14,   41,  137,   491,   1829,  ...

%e   10,  28,  107,  487,  2429,  12763,  ...

%t Table[Function[k, SeriesCoefficient[Product[(1 + x^j)^DivisorSigma[k, j], {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 10}, {n, 0, i}] // Flatten

%t Table[Function[k, SeriesCoefficient[Exp[Sum[DivisorSigma[k + 1, j] x^j/(j (1 - x^(2 j))), {j, 1, n}]], {x, 0, n}]][i - n], {i, 0, 10}, {n, 0, i}] // Flatten

%Y Columns k=0..9 give A107742, A192065, A288414, A288415, A301548, A301549, A301550, A301551, A301552, A301553.

%Y Main diagonal gives A321042.

%Y Cf. A321876.

%K nonn,tabl

%O 0,6

%A _Ilya Gutkovskiy_, Nov 20 2018