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A303174 a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n-k+1).

%I A303174 #9 Aug 21 2018 10:53:14
%S A303174 1,-1,2,-5,18,-60,189,-601,1967,-6544,21872,-73247,246080,-829924,
%T A303174 2808357,-9527485,32389671,-110316862,376372802,-1286063899,
%U A303174 4400499380,-15075608840,51704898623,-177513230200,610007283817,-2098029341745,7221561430933,-24875274224531
%N A303174 a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n-k+1).
%H A303174 Vaclav Kotesovec, <a href="/A303174/b303174.txt">Table of n, a(n) for n = 0..500</a>
%F A303174 a(n) ~ (-1)^n * c * d^n / sqrt(n), where d = A318204 = 3.50975432794970334043727352337... and c = 0.2457469629428839220188283... - _Vaclav Kotesovec_, Aug 21 2018
%e A303174 a(0) = 1;
%e A303174 a(1) = [x^1] 1/(1 + x) = -1;
%e A303174 a(2) = [x^2] 1/((1 + x)^2*(1 + x^2)) = 2;
%e A303174 a(3) = [x^3] 1/((1 + x)^3*(1 + x^2)^2*(1 + x^3)) = -5;
%e A303174 a(4) = [x^4] 1/((1 + x)^4*(1 + x^2)^3*(1 + x^3)^2*(1 + x^4)) = 18;
%e A303174 a(5) = [x^5] 1/((1 + x)^5*(1 + x^2)^4*(1 + x^3)^3*(1 + x^4)^2*(1 + x^5)) = -60, etc.
%e A303174 ...
%e A303174 The table of coefficients of x^k in expansion of Product_{k=1..n} 1/(1 + x^k)^(n-k+1) begins:
%e A303174 n = 0: (1),  0,   0,    0,   0,    0,  ...
%e A303174 n = 1:  1, (-1),  1,   -1,   1,   -1,  ...
%e A303174 n = 2:  1,  -2,  (2),  -2,   3,   -4,  ...
%e A303174 n = 3:  1,  -3,   4,  (-5),  9,  -14,  ...
%e A303174 n = 4:  1,  -4,   7,  -10, (18), -30,  ...
%e A303174 n = 5:  1,  -5,  11,  -18,  33, (-60), ...
%t A303174 Table[SeriesCoefficient[Product[1/(1 + x^k)^(n - k + 1), {k, 1, n}], {x, 0, n}], {n, 0, 27}]
%Y A303174 Cf. A206228, A206229, A255526, A255528, A303173.
%K A303174 sign
%O A303174 0,3
%A A303174 _Ilya Gutkovskiy_, Apr 19 2018