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

1, -1, 0, 4, -7, 0, 13, 10, -92, 21, 720, -2019, 1193, 6281, -18054, 16111, 11059, -14653, -57685, -86620, 1281406, -3454742, 2383734, 9409968, -30397071, 43327680, -56130326, 128981571, -73487834, -1219918457, 5059678044, -7826243881, -4131571113, 38850603452

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 - x)^2*(1 - x^2) = 0;
a(3) = [x^3] (1 - x)^3*(1 - x^2)^2*(1 - x^3) = 4;
a(4) = [x^4] (1 - x)^4*(1 - x^2)^3*(1 - x^3)^2*(1 - x^4) = -7;
a(5) = [x^5] (1 - x)^5*(1 - x^2)^4*(1 - x^3)^3*(1 - x^4)^2*(1 - x^5) = 0, etc.
...
The table of coefficients of x^k in expansion of Product_{k=1..n} (1 - x^k)^(n-k+1) begins:
n = 0: (1),  0,  0,  0,    0,   0,  ...
n = 1:  1, (-1), 0,  0,    0,   0,  ...
n = 2:  1,  -2, (0), 2,   -1,   0,  ...
n = 3:  1,  -3,  1, (4),  -2,  -2,  ...
n = 4:  1,  -4,  3,  6,  (-7), -2,  ...
n = 5:  1,  -5,  6,  7,  -16,  (0), ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A008705, A073592, A206228, A206229, A303174.

Programs

Mathematica

Table[SeriesCoefficient[Product[(1 - x^k)^(n - k + 1), {k, 1, n}], {x, 0, n}], {n, 0, 33}]