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A192890 Expansion of g.f.: exp( Sum_{n>=1} 2^n*(Sum_{d|n} d*x^d)^n/n ). 3

%I

%S 1,2,4,16,40,136,400,1256,4272,14984,51856,174304,563776,1820672,

%T 5780864,18518912,61112832,213235904,775157504,2859529984,10441532416,

%U 37338042528,130397329216,447932053120,1524595536512,5194654080416,17800611666496

%N Expansion of g.f.: exp( Sum_{n>=1} 2^n*(Sum_{d|n} d*x^d)^n/n ).

%H G. C. Greubel, <a href="/A192890/b192890.txt">Table of n, a(n) for n = 0..1000</a>

%e G.f.: A(x) = 1 + 2*x + 4*x^2 + 16*x^3 + 40*x^4 + 136*x^5 + 400*x^6 + 1256*x^7 +...

%e where the logarithm of the g.f. begins:

%e log(A(x)) = 2*x + 4*(x + 2*x^2)^2/2 + 8*(x + 3*x^3)^3/3 + 16*(x + 2*x^2 + 4*x^4)^4/4 + 32*(x + 5*x^5)^5/5 + 64*(x + 2*x^2 + 3*x^3 + 6*x^6)^6/6 + 128*(x + 7*x^7)^7/7 + 256*(x + 2*x^2 + 4*x^4 + 8*x^8)^8/8 +...

%e Explicitly, the logarithmic series begins:

%e log(A(x)) = 2*x + 4*x^2/2 + 32*x^3/3 + 48*x^4/4 + 312*x^5/5 + 640*x^6/6 + 2872*x^7/7 + 10496*x^8/8 + 46760*x^9/9 + 162624*x^10/10 +...

%t With[{m = 30}, CoefficientList[Series[Exp[Sum[2^n (Sum[d*x^d, {d, Divisors[n]}])^n/n, {n, 1, m + 2}]], {x, 0, m}], x]] (* _G. C. Greubel_, Jan 09 2019 *)

%o (PARI) {a(n)=local(A);A=exp(sum(m=1,n+1,2^m*sumdiv(m,d,d*x^d +x*O(x^n))^m/m));polcoeff(A,n)}

%Y Cf. A192860, A192891.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 11 2011

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Last modified October 21 22:47 EDT 2021. Contains 348160 sequences. (Running on oeis4.)