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

1, -1, -1, 5, 7, 21, -94, -117, -404, -840, 3541, 4536, 14412, 31313, 72175, -249424, -262828, -930639, -1895460, -4441316, -8085972, 24112570, 26214408, 87131883, 180197979, 411759028, 748154122, 1525043990, -3554837744, -3210408245, -11955482059, -23817949142, -55221348072

Offset: 0

Author

Ilya Gutkovskiy, Apr 19 2018

Keywords

• sign

Examples

			a(0) = 1;
a(1) = [x^1] (1 - x) = -1;
a(2) = [x^2] (1 - 2*x)*(1 - x^2) = -1;
a(3) = [x^3] (1 - 3*x)*(1 - 2*x^2)*(1 - x^3) = 5;
a(4) = [x^4] (1 - 4*x)*(1 - 3*x^2)*(1 - 2*x^3)*(1 - x^4) = 7;
a(5) = [x^5] (1 - 5*x)*(1 - 4*x^2)*(1 - 3*x^3)*(1 - 2*x^4)*(1 - x^5) = 21, etc.
...
The table of coefficients of x^k in expansion of Product_{k=1..n} (1 - (n - k + 1)*x^k) begins:
n = 0: (1),  0,   0,   0,   0,   0,  ...
n = 1:  1, (-1),  0,   0,   0,   0,  ...
n = 2:  1,  -2, (-1),  2,   0,   0   ...
n = 3:  1,  -3,  -2,  (5),  3,   2,  ...
n = 4:  1,  -4,  -3,  10,  (7), 10,  ...
n = 5:  1,  -5,  -4,  17,  13, (21), ...
		

Crossrefs

Cf. A022661, A292132, A303173, A303175, A303188, A303190.

Programs

Mathematica

Table[SeriesCoefficient[Product[(1 - (n - k + 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 32}]