The OEIS is supported by the many generous donors to the OEIS Foundation.
Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #10 Dec 06 2020 06:27:14
%S 1,1,-1,1,0,-1,1,1,0,0,1,2,3,0,0,1,3,8,6,0,1,1,4,15,24,14,0,0,1,5,24,
%T 60,78,27,0,1,1,6,35,120,252,232,60,0,0,1,7,48,210,620,1005,720,117,0,
%U 0,1,8,63,336,1290,3096,4080,2152,246,0,0,1,9,80,504,2394,7735,15600,16305,6528,490,0,0
%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)/(1 - k*x^j).
%H Seiichi Manyama, <a href="/A319753/b319753.txt">Antidiagonals n = 0..139, flattened</a>
%F G.f. of column k: Product_{j>=1} (1 - x^j)/(1 - k*x^j).
%F G.f. of column k: exp(Sum_{j>=1} ( Sum_{d|j} d*(k^(j/d) - 1) ) * x^j/j).
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e -1, 0, 1, 2, 3, 4, ...
%e -1, 0, 3, 8, 15, 24, ...
%e 0, 0, 6, 24, 60, 120, ...
%e 0, 0, 14, 78, 252, 620, ...
%e 1, 0, 27, 232, 1005, 3096, ...
%t Table[Function[k, SeriesCoefficient[Product[(1 - x^i)/(1 - k x^i), {i, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%t Table[Function[k, SeriesCoefficient[Exp[Sum[Sum[d (k^(i/d) - 1), {d, Divisors[i]}] x^i/i, {i, n}]], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%Y Columns k=0..20 give A010815, A000007, A006951, A006952, A049314, A049315, A221578, A049316, A182603, A182604, A221579, A182605, A221580, A182606, A221581, A221582, A182607, A182608, A221583, A182609, A221584.
%Y Main diagonal gives A319754.
%K sign,tabl
%O 0,12
%A _Ilya Gutkovskiy_, Sep 27 2018