A158102 - OEIS

%I #5 Nov 24 2015 03:38:41

%S 1,2,14,192,6460,604352,176638592,175532923904,624721970104800,

%T 8175475971987294720,397644417676499319116800,

%U 72130520531109535961648537600,48849638890950023005189368829890560

%N a(n) = [x^n] 1/eta(x)^(2^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(2^n*x) )^n/n!.

%F G.f.: A(x) = Sum_{n>=0} [ Sum_{k>=1} ( (2^n*x)^k/(1 - (2^n*x)^k) )/k ]^n/n!.

%F a(n) = [x^n] P(x)^(2^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 + 2*x + 14*x^2 + 192*x^3 + 6460*x^4 + 604352*x^5 +...

%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)^(2^n):

%e P(x)^(2^0): [(1),1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,...];

%e P(x)^(2^1): [1,(2),5,10,20,36,65,110,185,300,481,752,1165,...];

%e P(x)^(2^2): [1,4,(14),40,105,252,574,1240,2580,5180,10108,...];

%e P(x)^(2^3): [1,8,44,(192),726,2464,7704,22528,62337,164560,...];

%e P(x)^(2^4): [1,16,152,1088,(6460),33440,155584,663936,...];

%e P(x)^(2^5): [1,32,560,7040,70840,(604352),4528832,30529280,...];

%e P(x)^(2^6): [1,64,2144,49920,905840,13627264,(176638592),...]; ...

%e where terms in parenthesis form the initial terms of this sequence.

%t a[n_] := SeriesCoefficient[1/QPochhammer[q]^(2^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))^(2^n), n)}

%o (PARI) {a(n)=polcoeff(sum(m=0,n,(-1)^m*log(eta(2^m*x+x*O(x^n)))^m/m!), n)}

%o (PARI) {a(n)=polcoeff(sum(m=0,n,sum(k=1,n,(2^m*x)^k/(1-(2^m*x)^k)/k+x*O(x^n))^m/m!),n)}

%Y Cf. A000041, A158103, A158104, A158105, A158112, A158113, A158114, A158115.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 12 2009