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A158105 a(n) = [x^n] 1/eta(x)^(5^n).

%I A158105 #5 Nov 24 2015 03:34:46
%S A158105 1,5,350,349125,6541895625,2507423437503750,20269247819848681671875,
%T A158105 3527360416889444660042550312500,
%U A158105 13447858828542523681813067677875519531250
%N A158105 a(n) = [x^n] 1/eta(x)^(5^n).
%C A158105 Here eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).
%F A158105 G.f.: A(x) = Sum_{n>=0} (-1)^n*log( eta(5^n*x) )^n/n!.
%F A158105 G.f.: A(x) = Sum_{n>=0} [ Sum_{k>=1} ( (5^n*x)^k/(1 - (5^n*x)^k) )/k ]^n/n!.
%F A158105 a(n) = [x^n] P(x)^(5^n) where P(x) = 1/eta(x) = Product_{n>0} 1/(1-x^n) = g.f. of the partition numbers (A000041).
%e A158105 G.f.: A(x) = 1 + 5*x + 350*x^2 + 349125*x^3 + 6541895625*x^4 +...
%e A158105 A(x) = 1 - log(eta(5*x)) + log(eta(25*x))^2/2! - log(eta(125*x))^3/3! +-...
%e A158105 ...
%e A158105 Let P(x) = 1/eta(x) denote the g.f. of the partition numbers A000041:
%e A158105 P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 +...
%e A158105 then a(n) is the coefficient of x^n in P(x)^(5^n):
%e A158105 P(x)^(5^0): [(1),1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,...];
%e A158105 P(x)^(5^1): [1,(5),20,65,190,506,1265,2990,6765,14725,31027,...];
%e A158105 P(x)^(5^2): [1,25,(350),3575,29575,209405,1312675,7452225,...];
%e A158105 P(x)^(5^3): [1,125,8000,(349125),11676000,318906400,...];
%e A158105 P(x)^(5^4): [1,625,196250,41276875,(6541895625),833314453875,...];
%e A158105 P(x)^(5^5): [1,3125,4887500,5100915625,3996555181250,(2507423437503750),..];
%e A158105 where terms in parenthesis form the initial terms of this sequence.
%t A158105 a[n_] := SeriesCoefficient[1/QPochhammer[q]^(5^n), {q, 0, n}]; Table[a[n], {n, 0, 10}] (* _Jean-François Alcover_, Nov 24 2015 *)
%o A158105 (PARI) {a(n)=polcoeff(1/eta(x+x*O(x^n))^(5^n), n)}
%o A158105 (PARI) {a(n)=polcoeff(sum(m=0,n,(-1)^m*log(eta(5^m*x+x*O(x^n)))^m/m!), n)}
%o A158105 (PARI) {a(n)=polcoeff(sum(m=0,n,sum(k=1,n,(5^m*x)^k/(1-(5^m*x)^k)/k+x*O(x^n))^m/m!),n)}
%Y A158105 Cf. A000041, A158102, A158103, A158104, A158112, A158113, A158114, A158115.
%K A158105 nonn
%O A158105 0,2
%A A158105 _Paul D. Hanna_, Mar 12 2009