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A286354 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)^k.

%I A286354 #18 Jun 21 2018 18:38:24
%S A286354 1,1,0,1,-1,0,1,-2,-1,0,1,-3,-1,0,0,1,-4,0,2,0,0,1,-5,2,5,1,1,0,1,-6,
%T A286354 5,8,0,2,0,0,1,-7,9,10,-5,0,-2,1,0,1,-8,14,10,-15,-4,-7,0,0,0,1,-9,20,
%U A286354 7,-30,-6,-10,0,-2,0,0,1,-10,27,0,-49,0,-5,8,0,-2,0,0,1,-11,35,-12,-70,21,11,25,9,0,1,0,0
%N A286354 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)^k.
%C A286354 A(n,k) number of partitions of n into an even number of distinct parts minus number of partitions of n into an odd number of distinct parts with k types of each part.
%H A286354 Seiichi Manyama, <a href="/A286354/b286354.txt">Antidiagonals n = 0..139, flattened</a>
%H A286354 <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>
%F A286354 G.f. of column k: Product_{j>=1} (1 - x^j)^k.
%F A286354 G.f. of column k: (Sum_{j=-inf..inf} (-1)^j*x^(j*(3*j+1)/2))^k.
%F A286354 Column k is the Euler transform of period 1 sequence [-k, -k, -k, ...].
%e A286354 A(3,2) = 2 because we have [2, 1], [2', 1], [2, 1'], [2', 1'] (number of partitions of 3 into an even number of distinct parts with 2 types of each part), [3], [3'] (number of partitions of 3 into an odd number of distinct parts with 2 types of each part) and 4 - 2 = 2.
%e A286354 Square array begins:
%e A286354 1,  1,  1,  1,  1,   1,  ...
%e A286354 0, -1, -2, -3, -4,  -5,  ...
%e A286354 0, -1, -1,  0,  2,   5,  ...
%e A286354 0,  0,  2,  5,  8,  10,  ...
%e A286354 0,  0,  1,  0, -5, -15,  ...
%e A286354 0,  1,  2,  0, -4,  -6,  ...
%p A286354 A:= proc(n, k) option remember; `if`(n=0, 1, -k*
%p A286354       add(numtheory[sigma](j)*A(n-j, k), j=1..n)/n)
%p A286354     end:
%p A286354 seq(seq(A(n, d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Jun 21 2018
%t A286354 Table[Function[k, SeriesCoefficient[Product[(1 - x^i)^k , {i, Infinity}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
%t A286354 Table[Function[k, SeriesCoefficient[QPochhammer[x, x, Infinity]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
%t A286354 Table[Function[k, SeriesCoefficient[Sum[(-1)^i*x^(i*(3*i + 1)/2), {i, -Infinity, Infinity}]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
%Y A286354 Columns k=0-20 give: A000007, A010815, A002107, A010816, A000727, A000728, A000729, A000730, A000731, A010817, A010818, A010819, A000735, A010820, A010821, A010822, A000739, A010823, A010824, A010825, A010826.
%Y A286354 Main diagonal gives A008705.
%Y A286354 Antidiagonal sums give A299105.
%K A286354 sign,tabl
%O A286354 0,8
%A A286354 _Ilya Gutkovskiy_, May 08 2017