%I #8 Aug 22 2018 06:12:54

%S 1,-1,3,-22,224,-2759,41629,-743319,15285861,-355719616,9242332881,

%T -265191971970,8328195163545,-284124989856012,10463788330880961,

%U -413744821089831397,17482192791456272614,-786119610413822514764,37482612103603819839034,-1888918995730788198553380

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

%H Vaclav Kotesovec, <a href="/A303190/b303190.txt">Table of n, a(n) for n = 0..384</a>

%F a(n) ~ (-1)^n * n^n * (1 - 1/n + 3/n^2 - 7/n^3 + 15/n^4 - 32/n^5 + 65/n^6 - 131/n^7 + 260/n^8 - 501/n^9 + 965/n^10 - 1825/n^11 + 3419/n^12 - 6326/n^13 + 11652/n^14 - 21230/n^15 + 38405/n^16 - 69015/n^17 + 123334/n^18 - 218980/n^19 + 386809/n^20 - 679757/n^21 + 1189360/n^22 - 2071761/n^23 + 3594325/n^24 - 6211826/n^25 + 10698409/n^26 - 18363038/n^27 + 31420994/n^28 - 53605525/n^29 + 91198970/n^30 - ...). - _Vaclav Kotesovec_, Aug 22 2018

%e a(0) = 1;

%e a(1) = [x^1] 1/(1 + x) = -1;

%e a(2) = [x^2] 1/((1 + 2*x)*(1 + x^2)) = 3;

%e a(3) = [x^3] 1/((1 + 3*x)*(1 + 2*x^2)*(1 + x^3)) = -22;

%e a(4) = [x^4] 1/((1 + 4*x)*(1 + 3*x^2)*(1 + 2*x^3)*(1 + x^4)) = 224;

%e a(5) = [x^5] 1/((1 + 5*x)*(1 + 4*x^2)*(1 + 3*x^3)*(1 + 2*x^4)*(1 + x^5)) = -2759, etc.

%e ...

%e The table of coefficients of x^k in expansion of Product_{k=1..n} 1/(1 + (n - k + 1)*x^k) begins:

%e n = 0: (1), 0, 0, 0, 0, 0, ...

%e n = 1: 1, (-1), 1, -1, 1, -1, ...

%e n = 2: 1, -2, (3), -6, 13, -26, ...

%e n = 3: 1, -3, 7, (-22), 70, -208, ...

%e n = 4: 1, -4, 13, -54, (224), -890, ...

%e n = 5: 1, -5, 21, -108, 554, (-2759), ...

%t Table[SeriesCoefficient[Product[1/(1 + (n - k + 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]

%Y Cf. A022693, A292134, A303174, A303175, A303188, A303189.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Apr 19 2018