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A303173 a(n) = [x^n] Product_{k=1..n} (1 - x^k)^(n-k+1).

%I A303173 #7 Aug 21 2018 07:41:40
%S A303173 1,-1,0,4,-7,0,13,10,-92,21,720,-2019,1193,6281,-18054,16111,11059,
%T A303173 -14653,-57685,-86620,1281406,-3454742,2383734,9409968,-30397071,
%U A303173 43327680,-56130326,128981571,-73487834,-1219918457,5059678044,-7826243881,-4131571113,38850603452
%N A303173 a(n) = [x^n] Product_{k=1..n} (1 - x^k)^(n-k+1).
%H A303173 Vaclav Kotesovec, <a href="/A303173/b303173.txt">Table of n, a(n) for n = 0..500</a>
%e A303173 a(0) = 1;
%e A303173 a(1) = [x^1] (1 - x) = -1;
%e A303173 a(2) = [x^2] (1 - x)^2*(1 - x^2) = 0;
%e A303173 a(3) = [x^3] (1 - x)^3*(1 - x^2)^2*(1 - x^3) = 4;
%e A303173 a(4) = [x^4] (1 - x)^4*(1 - x^2)^3*(1 - x^3)^2*(1 - x^4) = -7;
%e A303173 a(5) = [x^5] (1 - x)^5*(1 - x^2)^4*(1 - x^3)^3*(1 - x^4)^2*(1 - x^5) = 0, etc.
%e A303173 ...
%e A303173 The table of coefficients of x^k in expansion of Product_{k=1..n} (1 - x^k)^(n-k+1) begins:
%e A303173 n = 0: (1),  0,  0,  0,    0,   0,  ...
%e A303173 n = 1:  1, (-1), 0,  0,    0,   0,  ...
%e A303173 n = 2:  1,  -2, (0), 2,   -1,   0,  ...
%e A303173 n = 3:  1,  -3,  1, (4),  -2,  -2,  ...
%e A303173 n = 4:  1,  -4,  3,  6,  (-7), -2,  ...
%e A303173 n = 5:  1,  -5,  6,  7,  -16,  (0), ...
%t A303173 Table[SeriesCoefficient[Product[(1 - x^k)^(n - k + 1), {k, 1, n}], {x, 0, n}], {n, 0, 33}]
%Y A303173 Cf. A008705, A073592, A206228, A206229, A303174.
%K A303173 sign
%O A303173 0,4
%A A303173 _Ilya Gutkovskiy_, Apr 19 2018