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A192731 Euler transform is 1 / (q j(q)) where j is j-function (A000521). 19

%I

%S -744,80256,-12288744,2126816256,-392642298600,75506620496256,

%T -14935073808384744,3015675387953504256,-618587635244888064744,

%U 128473308888136855075200,-26951900214112779571200744

%N Euler transform is 1 / (q j(q)) where j is j-function (A000521).

%H Seiichi Manyama, <a href="/A192731/b192731.txt">Table of n, a(n) for n = 1..424</a>

%H B. Brent, <a href="http://mathoverflow.net/questions/69365/">p-adic continuity for exponents in product decomposition of the j-invariant</a>, Answer 3 by W. Zudilin

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F 1 / (q j(q)) = Product_{k>0} (1 - x^k)^-a(k).

%F a(n) = 3*(A110163(n) - 8) = (1/n) * Sum_{d|n} A008683(n/d) * A288261(d). - _Seiichi Manyama_, Jun 18 2017

%F a(n) ~ (-1)^n * 3*exp(Pi*sqrt(3)*n) / n. - _Vaclav Kotesovec_, Mar 24 2018

%e From _Seiichi Manyama_, Jun 18 2017: (Start)

%e a(1) = (1/1) * A008683(1/1) * A288261(1) = (1/1) * (-744) = -744,

%e a(2) = (1/2) * (A008683(2/1) * A288261(1) + A008683(2/2) * A288261(2)) = (1/2) * (744 + 159768) = 80256. (End)

%o (PARI) {a(n) = local(A, S); if( n<1, 0, A = 1 + x * O(x^n); S = x * ellj( x * A ); for( k = 1, n-1, S *= (A - x^k) ^ polcoeff( S, k)); - polcoeff( S, n))}

%Y Cf. A008683, A063995, A110163, A192732, A288261, A302407, A305757.

%K sign

%O 1,1

%A _Michael Somos_, Jul 08 2011

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Last modified July 25 03:53 EDT 2021. Contains 346283 sequences. (Running on oeis4.)