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A303175 a(n) = [x^n] Product_{k=1..n} 1/(1 - (n - k + 1)*x^k).

%I A303175 #10 Aug 21 2018 07:46:27
%S A303175 1,1,5,34,322,3803,55297,953815,19086057,434477488,11086102633,
%T A303175 313318606066,9714265351819,327788649292844,11957321196905337,
%U A303175 468872400449456885,19666225828334583690,878560858388253803180,41645712575272737701666,2087686693048676581394052
%N A303175 a(n) = [x^n] Product_{k=1..n} 1/(1 - (n - k + 1)*x^k).
%H A303175 Vaclav Kotesovec, <a href="/A303175/b303175.txt">Table of n, a(n) for n = 0..384</a>
%F A303175 a(n) ~ n^n * (1 + 1/n + 1/n^2 - 1/n^3 - 3/n^4 - 8/n^5 - 7/n^6 - 13/n^7 + 2/n^8 - 3/n^9 + 31/n^10 + 21/n^11 + 81/n^12 + 2/n^13 + 152/n^14 - 114/n^15 + 173/n^16 - 341/n^17 + 260/n^18 - 936/n^19 + 861/n^20 - 2187/n^21 + 2630/n^22 - 4551/n^23 + 6211/n^24 - 8866/n^25 + 14889/n^26 - 22374/n^27 + 38490/n^28 - 55911/n^29 + 87688/n^30 - ...). - _Vaclav Kotesovec_, Aug 21 2018
%e A303175 a(0) = 1;
%e A303175 a(1) = [x^1] 1/(1 - x) = 1;
%e A303175 a(2) = [x^2] 1/((1 - 2*x)*(1 - x^2)) = 5;
%e A303175 a(3) = [x^3] 1/((1 - 3*x)*(1 - 2*x^2)*(1 - x^3)) = 34;
%e A303175 a(4) = [x^4] 1/((1 - 4*x)*(1 - 3*x^2)*(1 - 2*x^3)*(1 - x^4)) = 322;
%e A303175 a(5) = [x^5] 1/((1 - 5*x)*(1 - 4*x^2)*(1 - 3*x^3)*(1 - 2*x^4)*(1 - x^5)) = 3803, etc.
%e A303175 ...
%e A303175 The table of coefficients of x^k in expansion of Product_{k=1..n} 1/(1 - (n - k + 1)*x^k) begins:
%e A303175 n = 0: (1), 0,   0,    0,    0,     0,  ...
%e A303175 n = 1:  1, (1),  1,    1,    1,     1,  ...
%e A303175 n = 2:  1,  2,  (5),  10,   21,    42,  ...
%e A303175 n = 3:  1,  3,  11,  (34), 106,   320,  ...
%e A303175 n = 4:  1,  4,  19,   78, (322), 1294,  ...
%e A303175 n = 5:  1,  5,  29,  148,  758, (3803), ...
%t A303175 Table[SeriesCoefficient[Product[1/(1 - (n - k + 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]
%Y A303175 Cf. A006906, A124577, A206228, A303188, A303189, A303190.
%K A303175 nonn
%O A303175 0,3
%A A303175 _Ilya Gutkovskiy_, Apr 19 2018