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 A158112 a(n) = [x^n] eta(x)^(2^n). 8
 1, -2, 2, 0, 260, -72384, 40618368, -64586112000, 322969883412000, -5357637135507147264, 305435344239722874022912, -61394251001333873555321724928, 44347756583930343994689166159720448 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Here eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815). LINKS FORMULA G.f.: A(x) = Sum_{n>=0} log( eta(2^n*x) )^n/n!. G.f.: A(x) = Sum_{n>=0} [ -Sum_{k>=1} ( (2^n*x)^k/(1 - (2^n*x)^k) )/k ]^n/n!. a(n) = [x^n] Product_{k>=1} (1-x^k)^(2^n). EXAMPLE G.f.: A(x) = 1 - 2*x + 2*x^2 + 260*x^4 - 72384*x^5 +... A(x) = 1 + log(eta(2*x)) + log(eta(4*x))^2/2! + log(eta(8*x))^3/3! +... ... Given eta(x) = 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 +... then a(n) is the coefficient of x^n in eta(x)^(2^n): eta(x)^(2^0): [(1),-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,0,0,0,0,..]; eta(x)^(2^1): [1,(-2),-1,2,1,2,-2,0,-2,-2,1,0,0,2,3,-2,2,...]; eta(x)^(2^2): [1,-4,(2),8,-5,-4,-10,8,9,0,14,-16,-10,-4,0,-8,...]; eta(x)^(2^3): [1,-8,20,(0),-70,64,56,0,-125,-160,308,0,110,0,...]; eta(x)^(2^4): [1,-16,104,-320,(260),1248,-3712,1664,6890,...]; eta(x)^(2^5): [1,-32,464,-3968,21576,(-72384),109120,215296,...]; eta(x)^(2^6): [1,-64,1952,-37632,512400,-5207936,(40618368),...]; ... where terms in parenthesis form the initial terms of this sequence. MATHEMATICA a[n_] := SeriesCoefficient[QPochhammer[q]^(2^n), {q, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 24 2015 *) PROG (PARI) {a(n)=polcoeff(eta(x+x*O(x^n))^(2^n), n)} (PARI) {a(n)=polcoeff(sum(m=0, n, log(eta(2^m*x+x*O(x^n)))^m/m!), n)} (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)} CROSSREFS Cf. A010815, A158113, A158114, A158115, A158102, A158103, A158104, A158105. Sequence in context: A245687 A228617 A119836 * A219496 A163534 A255124 Adjacent sequences: A158109 A158110 A158111 * A158113 A158114 A158115 KEYWORD sign AUTHOR Paul D. Hanna, Mar 12 2009 STATUS approved

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Last modified March 27 17:48 EDT 2023. Contains 361575 sequences. (Running on oeis4.)