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

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A158103 a(n) = [x^n] 1/eta(x)^(3^n). 7
 1, 3, 54, 4410, 2208465, 7958364696, 221555929999779, 48859965926267395185, 86255750314864791590005098, 1228682270675324224826503933533795, 142349199783036538823503393789360721783250 (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} (-1)^n*log( eta(3^n*x) )^n/n!. G.f.: A(x) = Sum_{n>=0} [ Sum_{k>=1} ( (3^n*x)^k/(1 - (3^n*x)^k) )/k ]^n/n!. 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). EXAMPLE G.f.: A(x) = 1 + 3*x + 54*x^2 + 4410*x^3 + 2208465*x^4 +... A(x) = 1 - log(eta(3*x)) + log(eta(9*x))^2/2! - log(eta(27*x))^3/3! +-... ... Let P(x) = 1/eta(x) denote the g.f. of the partition numbers A000041: P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 +... then a(n) is the coefficient of x^n in P(x)^(3^n): P(x)^(3^0): [(1),1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,...]; P(x)^(3^1): [1,(3),9,22,51,108,221,429,810,1479,2640,4599,...]; P(x)^(3^2): [1,9,(54),255,1035,3753,12483,38709,113265,...]; P(x)^(3^3): [1,27,405,(4410),38745,290466,1923075,11506185,...]; P(x)^(3^4): [1,81,3402,98523,(2208465),40795083,645824907,...]; P(x)^(3^5): [1,243,29889,2480382,156189951,(7958364696),...]; where terms in parenthesis form the initial terms of this sequence. MATHEMATICA a[n_] := SeriesCoefficient[1/QPochhammer[q]^(3^n), {q, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 24 2015 *) PROG (PARI) {a(n)=polcoeff(1/eta(x+x*O(x^n))^(3^n), n)} (PARI) {a(n)=polcoeff(sum(m=0, n, (-1)^m*log(eta(3^m*x+x*O(x^n)))^m/m!), n)} (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)} CROSSREFS Cf. A000041, A158102, A158104, A158105, A158112, A158113, A158114, A158115. Sequence in context: A049414 A003027 A054545 * A174579 A171739 A157568 Adjacent sequences: A158100 A158101 A158102 * A158104 A158105 A158106 KEYWORD nonn 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.

Last modified December 4 21:02 EST 2022. Contains 358570 sequences. (Running on oeis4.)