Address: [go: up one dir, main page]

---

Include Form Remove Scripts Session Cookies

login

The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!) A158114 a(n) = [x^n] eta(x)^(4^n). 7 1, -4, 104, -37632, 166534720, -9109541173248, 6487005386806124544, -62637995710787181892993024, 8428730138560436521519921925857280, -16103390694987849639716307556519680725483520 (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 Table of n, a(n) for n=0..9. FORMULA G.f.: A(x) = Sum_{n>=0} log( eta(4^n*x) )^n/n!. G.f.: A(x) = Sum_{n>=0} [ -Sum_{k>=1} ( (4^n*x)^k/(1 - (4^n*x)^k) )/k ]^n/n!. a(n) = [x^n] Product_{k>=1} (1-x^k)^(4^n). EXAMPLE G.f.: A(x) = 1 - 4*x + 104*x^2 - 37632*x^3 + 166534720*x^4 +... A(x) = 1 + log(eta(4*x)) + log(eta(16*x))^2/2! + log(eta(64*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)^(4^n): eta(x)^(4^0): [(1),-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,0,0,0,0,..]; eta(x)^(4^1): [1,(-4),2,8,-5,-4,-10,8,9,0,14,-16,-10,-4,0,-8,...]; eta(x)^(4^2): [1,-16,(104),-320,260,1248,-3712,1664,6890,...]; eta(x)^(4^3): [1,-64,1952,(-37632),512400,-5207936,40618368,...]; eta(x)^(4^4): [1,-256,32384,-2698240,(166534720),-8118668800,...]; eta(x)^(4^5): [1,-1024,522752,-177385472,45010254080,(-9109541173248), ...]; where terms in parenthesis form the initial terms of this sequence. PROG (PARI) {a(n)=polcoeff(eta(x+x*O(x^n))^(4^n), n)} (PARI) {a(n)=polcoeff(sum(m=0, n, log(eta(4^m*x+x*O(x^n)))^m/m!), n)} (PARI) {a(n)=polcoeff(sum(m=0, n, sum(k=1, n, -(4^m*x)^k/(1-(4^m*x)^k)/k+x*O(x^n))^m/m!), n)} CROSSREFS Cf. A010815, A158112, A158113, A158115, A158102, A158103, A158104, A158105. Sequence in context: A013114 A281430 A354174 * A082736 A157039 A171117 Adjacent sequences: A158111 A158112 A158113 * A158115 A158116 A158117 KEYWORD sign AUTHOR Paul D. Hanna, Mar 12 2009 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 28 01:15 EDT 2024. Contains 375477 sequences. (Running on oeis4.)