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A303189 a(n) = [x^n] Product_{k=1..n} (1 - (n - k + 1)*x^k).

%I A303189 #6 Apr 19 2018 15:13:39
%S A303189 1,-1,-1,5,7,21,-94,-117,-404,-840,3541,4536,14412,31313,72175,
%T A303189 -249424,-262828,-930639,-1895460,-4441316,-8085972,24112570,26214408,
%U A303189 87131883,180197979,411759028,748154122,1525043990,-3554837744,-3210408245,-11955482059,-23817949142,-55221348072
%N A303189 a(n) = [x^n] Product_{k=1..n} (1 - (n - k + 1)*x^k).
%e A303189 a(0) = 1;
%e A303189 a(1) = [x^1] (1 - x) = -1;
%e A303189 a(2) = [x^2] (1 - 2*x)*(1 - x^2) = -1;
%e A303189 a(3) = [x^3] (1 - 3*x)*(1 - 2*x^2)*(1 - x^3) = 5;
%e A303189 a(4) = [x^4] (1 - 4*x)*(1 - 3*x^2)*(1 - 2*x^3)*(1 - x^4) = 7;
%e A303189 a(5) = [x^5] (1 - 5*x)*(1 - 4*x^2)*(1 - 3*x^3)*(1 - 2*x^4)*(1 - x^5) = 21, etc.
%e A303189 ...
%e A303189 The table of coefficients of x^k in expansion of Product_{k=1..n} (1 - (n - k + 1)*x^k) begins:
%e A303189 n = 0: (1),  0,   0,   0,   0,   0,  ...
%e A303189 n = 1:  1, (-1),  0,   0,   0,   0,  ...
%e A303189 n = 2:  1,  -2, (-1),  2,   0,   0   ...
%e A303189 n = 3:  1,  -3,  -2,  (5),  3,   2,  ...
%e A303189 n = 4:  1,  -4,  -3,  10,  (7), 10,  ...
%e A303189 n = 5:  1,  -5,  -4,  17,  13, (21), ...
%t A303189 Table[SeriesCoefficient[Product[(1 - (n - k + 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 32}]
%Y A303189 Cf. A022661, A292132, A303173, A303175, A303188, A303190.
%K A303189 sign
%O A303189 0,4
%A A303189 _Ilya Gutkovskiy_, Apr 19 2018