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%I #6 Nov 24 2015 04:37:52
%S 1,3,54,4410,2208465,7958364696,221555929999779,48859965926267395185,
%T 86255750314864791590005098,1228682270675324224826503933533795,
%U 142349199783036538823503393789360721783250
%N a(n) = [x^n] 1/eta(x)^(3^n).
%C Here eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).
%F G.f.: A(x) = Sum_{n>=0} (-1)^n*log( eta(3^n*x) )^n/n!.
%F G.f.: A(x) = Sum_{n>=0} [ Sum_{k>=1} ( (3^n*x)^k/(1 - (3^n*x)^k) )/k ]^n/n!.
%F a(n) = [x^n] P(x)^(3^n) where P(x) = 1/eta(x) = Product_{n>0} 1/(1-x^n) = g.f. of the partition numbers (A000041).
%e G.f.: A(x) = 1 + 3*x + 54*x^2 + 4410*x^3 + 2208465*x^4 +...
%e A(x) = 1 - log(eta(3*x)) + log(eta(9*x))^2/2! - log(eta(27*x))^3/3! +-...
%e ...
%e Let P(x) = 1/eta(x) denote the g.f. of the partition numbers A000041:
%e P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 +...
%e then a(n) is the coefficient of x^n in P(x)^(3^n):
%e P(x)^(3^0): [(1),1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,...];
%e P(x)^(3^1): [1,(3),9,22,51,108,221,429,810,1479,2640,4599,...];
%e P(x)^(3^2): [1,9,(54),255,1035,3753,12483,38709,113265,...];
%e P(x)^(3^3): [1,27,405,(4410),38745,290466,1923075,11506185,...];
%e P(x)^(3^4): [1,81,3402,98523,(2208465),40795083,645824907,...];
%e P(x)^(3^5): [1,243,29889,2480382,156189951,(7958364696),...];
%e where terms in parenthesis form the initial terms of this sequence.
%t a[n_] := SeriesCoefficient[1/QPochhammer[q]^(3^n), {q, 0, n}]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Nov 24 2015 *)
%o (PARI) {a(n)=polcoeff(1/eta(x+x*O(x^n))^(3^n), n)}
%o (PARI) {a(n)=polcoeff(sum(m=0,n,(-1)^m*log(eta(3^m*x+x*O(x^n)))^m/m!), n)}
%o (PARI) {a(n)=polcoeff(sum(m=0,n,sum(k=1,n,(3^m*x)^k/(1-(3^m*x)^k)/k+x*O(x^n))^m/m!),n)}
%Y Cf. A000041, A158102, A158104, A158105, A158112, A158113, A158114, A158115.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Mar 12 2009
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