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A166869 a(n) = n * A056219(n+1).

%I A166869 #14 Sep 08 2022 08:45:48
%S A166869 2,4,12,20,30,54,91,120,171,250,374,504,663,854,1170,1568,2074,2628,
%T A166869 3325,4180,5313,6754,8602,10656,13100,16042,19683,24024,29464,36000,
%U A166869 43834,52768,63228,75582,90510,107856,128575,153178,182208,215400
%N A166869 a(n) = n * A056219(n+1).
%H A166869 G. C. Greubel, <a href="/A166869/b166869.txt">Table of n, a(n) for n = 1..1000</a>
%p A166869 N:= 100; b:= seq(coeff(series(add(x^((1/2)*n*(n+1))*mul(x +1/(1-x^k), k=1..n), n = 1..floor((1/2)*sqrt(9+8*N))), x, N+2), x, j), j = 1..N+1); seq(n*b[n+1], n=1..N); # _G. C. Greubel_, Nov 29 2019
%t A166869 max:= 100; b:= CoefficientList[Series[Sum[x^(n*(n+1)/2)*Product[(x +1/(1-x^k)), {k, n}], {n, Floor[Sqrt[9 +8*(max+5)]/2]}], {x, 0, max+5}], x]; Table[n*b[[n + 2]], {n, max}] (* _G. C. Greubel_, Nov 29 2019 *)
%o A166869 (Magma)
%o A166869 max:=50;
%o A166869 R<x>:=PowerSeriesRing(Integers(), max); b:= Coefficients(R!( (&+[x^Binomial(n+1,2)*(&*[x + 1/(1-x^j): j in [1..n]]): n in [1..Floor(Sqrt(9+8*max)/2)]]) ));
%o A166869 [(n-1)*b[n]: n in [2..max-1]]; // _G. C. Greubel_, Nov 29 2019
%o A166869 (Sage)
%o A166869 max=50;
%o A166869 def A056219_list(prec):
%o A166869     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166869     return P( sum(x^binomial(n+1,2)*product((x + 1/(1-x^j)) for j in (1..n)) for n in (1..floor(sqrt(9+8*max)/2))) ).list()
%o A166869 b=A056219_list(max);
%o A166869 [(n-1)*b[n] for n in (2..max)] # _G. C. Greubel_, Nov 29 2019
%Y A166869 Cf. A056219, A166870.
%K A166869 nonn
%O A166869 1,1
%A A166869 _Roger L. Bagula_, Oct 22 2009