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%I #8 Aug 22 2023 07:59:43
%S 3,171,59076,172200087,4236443047458,900567812945319804,
%T 1675095304614322707132768,27469470562340107289343634221615,
%U 3990838394187339929534246698449141509871,5153775207320113310364614887146151186290992699106
%N a(n) = Sum_{d|n} d*3^(n*d).
%C Equals the logarithmic derivative of A209495.
%F L.g.f.: Sum_{n>=1} -log(1 - 3^(n^2)*x^n).
%F L.g.f.: log( Sum_{n>=0} 3^(n^2)*x^n / Product_{k=1..n} (1 - 3^(k^2)*x^k) ).
%e L.g.f.: L(x) = 3*x + 171*x^2/2 + 59076*x^3/3 + 172200087*x^4/4 + ...
%e where exponentiation yields the g.f. of A209495:
%e exp(L(x)) = 1 + 3*x + 90*x^2 + 19953*x^3 + 43113141*x^4 + ...
%t a[n_] := DivisorSum[n, # * 3^(n*#) &]; Array[a, 10] (* _Amiram Eldar_, Aug 22 2023 *)
%o (PARI) {a(n)=sumdiv(n,d,d*3^(n*d))}
%o (PARI) {a(n)=n*polcoeff(sum(m=1,n,-log(1 - 3^(m^2)*x^m +x*O(x^n))),n)}
%o (PARI) {a(n)=n*polcoeff(log(1+sum(m=1, n, 3^(m^2)*x^m/prod(k=1, m, 1-(3^k*x)^k+x*O(x^n)))), n)}
%o for(n=1, 15, print1(a(n), ", "))
%Y Cf. A209495, A209803.
%K nonn
%O 1,1
%A _Paul D. Hanna_, Mar 13 2012
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