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A192065 Expansion of prod(k>=1, Q(x^k)^k ) where Q(x)=prod(k>=1, 1+x^k ). 28

%I

%S 1,1,3,7,14,28,58,106,201,372,669,1187,2101,3624,6229,10591,17796,

%T 29659,49107,80492,131157,212237,341084,544883,865717,1367233,2148552,

%U 3359490,5227270,8096544,12486800,19174319,29326306,44678825,67811375,102549673,154545549

%N Expansion of prod(k>=1, Q(x^k)^k ) where Q(x)=prod(k>=1, 1+x^k ).

%C Euler transform of A002131. - _Vaclav Kotesovec_, Mar 26 2018

%H Seiichi Manyama, <a href="/A192065/b192065.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)

%F a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A288418(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Jun 09 2017

%F a(n) ~ exp(3*Pi^(2/3) * Zeta(3)^(1/3) * n^(2/3)/2^(5/3) - Pi^(4/3) * n^(1/3) / (3*2^(7/3) * Zeta(3)^(1/3)) - Pi^2 / (864 * Zeta(3))) * Zeta(3)^(1/6) / (2^(19/24) * sqrt(3) * Pi^(1/6) * n^(2/3)). - _Vaclav Kotesovec_, Mar 23 2018

%t nn = 30; b = Table[DivisorSigma[1, n], {n, nn}]; CoefficientList[Series[Product[(1 + x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* _T. D. Noe_, Jun 19 2012 *)

%t kmax = 37; Product[QPochhammer[-1, x^k]^k/2^k, {k, 1, kmax}] + O[x]^kmax // CoefficientList[#, x]& (* _Jean-Fran├žois Alcover_, Jul 03 2017 *)

%t nmax = 40; CoefficientList[Series[Exp[Sum[Sum[DivisorSum[k, # / GCD[#, 2] &] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 31 2018 *)

%o (PARI) N=66; x='x+O('x^N); /* that many terms */

%o Q(x)=prod(k=1,N,1+x^k);

%o gf=prod(k=1,N, Q(x^k)^k ); /* == 1 +x +3*x^2 +7*x^3 +14*x^4 +28*x^5 +58*x^6 +... */

%o Vec(gf) /* show terms */ /* _Joerg Arndt_, Jun 24 2011 */

%Y Cf. A061256 (1/prod(k>=1, P(x^k)^k ) where P(x)=prod(k>=1, 1-x^k ) ).

%Y Product_{k>=1} (1 + x^k)^sigma_m(k): A107742 (m=0), this sequence (m=1), A288414 (m=2), A288415 (m=3), A301548 (m=4), A301549 (m=5), A301550 (m=6), A301551 (m=7), A301552 (m=8).

%K nonn

%O 0,3

%A _Joerg Arndt_, Jun 24 2011

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Last modified July 8 03:25 EDT 2020. Contains 335503 sequences. (Running on oeis4.)