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A321947 Column k=2 of triangle A257673.

%I A321947 #20 Aug 23 2021 06:33:53
%S A321947 1,6,21,62,162,396,917,2036,4380,9152,18694,37380,73444,141918,270370,
%T A321947 508178,943876,1733468,3151396,5674152,10126435,17921016,31468623,
%U A321947 54848750,94935565,163232096,278903915,473693432,799949111,1343550666,2244807927,3731885232
%N A321947 Column k=2 of triangle A257673.
%H A321947 Alois P. Heinz, <a href="/A321947/b321947.txt">Table of n, a(n) for n = 2..10000</a>
%F A321947 G.f.: (-1 + Product_{k>=1} 1 / (1 - x^k)^k)^2. - _Ilya Gutkovskiy_, Jan 30 2021
%F A321947 a(n) = A161870(n) - 2*A000219(n). - _Vaclav Kotesovec_, Jan 30 2021
%p A321947 b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
%p A321947       b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
%p A321947     end:
%p A321947 a:= n-> (k-> add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(2):
%p A321947 seq(a(n), n=2..35);
%t A321947 A321947[n_] := Module[{nn = n}, SeriesCoefficient[Product[1/(1 - x^i)^(2 i), {i, 1, nn}], {x, 0, nn}] - 2*SeriesCoefficient[Product[(1 - x^k)^-k, {k, nn}], {x, 0, nn}]]; Table[A321947[n], {n, 2, 33}] (* _Robert P. P. McKone_, Jan 30 2021 *)
%t A321947 b[n_, k_] := b[n, k] = If[n == 0, 1, k*Sum[
%t A321947      b[n - j, k]*DivisorSigma[2, j], {j, 1, n}]/n];
%t A321947 a[n_] := With[{k = 2}, Sum[b[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]];
%t A321947 Table[a[n], {n, 2, 35}] (* _Jean-François Alcover_, Aug 23 2021, after _Alois P. Heinz_ *)
%Y A321947 Column k=2 of A257673.
%Y A321947 Cf. A000219, A161870.
%K A321947 nonn
%O A321947 2,2
%A A321947 _Alois P. Heinz_, Nov 22 2018