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A303188 a(n) = [x^n] Product_{k=1..n} (1 + (n - k + 1)*x^k).

%I A303188 #8 Aug 21 2018 08:04:48
%S A303188 1,1,1,7,9,23,148,221,526,1040,6767,9664,23456,43943,91363,499028,
%T A303188 736410,1650395,3107540,6210372,10819270,57864166,80663444,179915133,
%U A303188 324882691,640398244,1087149284,2039724322,9121580902,12913282685,27250167385,48645989650,92634730208,156124357449
%N A303188 a(n) = [x^n] Product_{k=1..n} (1 + (n - k + 1)*x^k).
%H A303188 Vaclav Kotesovec, <a href="/A303188/b303188.txt">Table of n, a(n) for n = 0..500</a>
%e A303188 a(0) = 1;
%e A303188 a(1) = [x^1] (1 + x) = 1;
%e A303188 a(2) = [x^2] (1 + 2*x)*(1 + x^2) = 1;
%e A303188 a(3) = [x^3] (1 + 3*x)*(1 + 2*x^2)*(1 + x^3) = 7;
%e A303188 a(4) = [x^4] (1 + 4*x)*(1 + 3*x^2)*(1 + 2*x^3)*(1 + x^4) = 9;
%e A303188 a(5) = [x^5] (1 + 5*x)*(1 + 4*x^2)*(1 + 3*x^3)*(1 + 2*x^4)*(1 + x^5) = 23, etc.
%e A303188 ...
%e A303188 The table of coefficients of x^k in expansion of Product_{k=1..n} (1 + (n - k + 1)*x^k) begins:
%e A303188 n = 0: (1), 0,  0,   0,   0,   0,  ...
%e A303188 n = 1:  1, (1), 0,   0,   0,   0,  ...
%e A303188 n = 2:  1,  2, (1),  2,   0,   0   ...
%e A303188 n = 3:  1,  3,  2,  (7),  3,   2,  ...
%e A303188 n = 4:  1,  4,  3,  14,  (9), 10,  ...
%e A303188 n = 5:  1,  5,  4,  23,  17, (23), ...
%t A303188 Table[SeriesCoefficient[Product[(1 + (n - k + 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 33}]
%Y A303188 Cf. A022629, A206229, A291698, A303175, A303189, A303190.
%K A303188 nonn
%O A303188 0,4
%A A303188 _Ilya Gutkovskiy_, Apr 19 2018