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A147843 a(n) = -n*A010815(n).

%I A147843 #40 Jan 19 2024 04:56:54
%S A147843 0,1,2,0,0,-5,0,-7,0,0,0,0,12,0,0,15,0,0,0,0,0,0,-22,0,0,0,-26,0,0,0,
%T A147843 0,0,0,0,0,35,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,-51,0,0,0,0,0,-57,0,0,0,
%U A147843 0,0,0,0,0,0,0,0,0,70,0,0,0,0,0,0,77,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-92,0,0,0,0,0,0,0,-100,0,0,0
%N A147843 a(n) = -n*A010815(n).
%C A147843 Convolved with the partition numbers A000041 = sigma(n) prefaced with a 0 gives (0, 1, 3, 4, 7, 6, 12, 8, 15, 13,...).
%C A147843 Triangle A174740 convolves the partition numbers with a variant of this sequence, having row sums = A000203, sigma(n). - _Gary W. Adamson_, Mar 28 2010
%H A147843 Seiichi Manyama, <a href="/A147843/b147843.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)
%F A147843 G.f.: -x * d eta(x)/dx (derivative)  where eta(x) = prod(n>=1, 1-x^n). - _Joerg Arndt_, Mar 14 2010
%F A147843 a(n) = Sum_{k=0..n-1} A010815(k)*sigma(n-k), where sigma(n) = A000203(n) is the sum of divisors of n. - _Paul D. Hanna_, Jul 02 2011
%e A147843 a(5) = -5 = (-5) * A010815(5) = (-5) * 1.
%t A147843 A010815[n_] := SeriesCoefficient[Product[1 - x^k, {k, n}], {x, 0, n}];
%t A147843 Table[-n*A010815[n], {n, 0, 50}] (* _G. C. Greubel_, Jun 13 2017 *)
%o A147843 (PARI) a(n) = -n * if(issquare(24*n + 1, &n), kronecker(12, n)); \\ _Amiram Eldar_, Jan 19 2024 after  _Michael Somos_ at A010815
%Y A147843 Cf. A010815, A000203, A000041, A001318, A174740.
%K A147843 sign,easy
%O A147843 0,3
%A A147843 _Gary W. Adamson_, Nov 15 2008