login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A163129 G.f.: A(q) = exp( Sum_{n>=1} sigma(n) * 3*A038500(n) * q^n/n ), where A038500(n) = highest power of 3 dividing n. 5
1, 3, 9, 30, 75, 180, 441, 969, 2070, 4431, 8964, 17775, 35094, 66975, 125865, 235053, 429096, 773766, 1386027, 2442372, 4260645, 7384578, 12640320, 21453975, 36192519, 60454713, 100250100, 165311094, 270391857, 439479198, 710631279 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

Define trisections by: A(q) = T_0(q) + T_1(q) + T_2(q), then:

3*T_0(q)/T_1(q) = 3*T_1(q)/T_2(q) = T9B(q), the g.f. of A058091,

which is the McKay-Thompson series of class 9B for Monster.

G.f.: 1/Product_{n>=0} R(q^(3^n))^(3^n) where R(q) = E(q)^3/E(q^3) and E(q) = Product_{k>=1} (1 - q^n). - Joerg Arndt, Aug 03 2011

EXAMPLE

G.f.: A(q) = 1 + 3*q + 9*q^2 + 30*q^3 + 75*q^4 + 180*q^5 + 441*q^6 + ...

log(A(q)) = 3*q + 9*q^2/2 + 36*q^3/3 + 21*q^4/4 + 18*q^5/5 + 108*q^6/6 + ...

Define TRISECTIONS:

T_0(q) = 1 + 30*q^3 + 441*q^6 + 4431*q^9 + 35094*q^12 + ...

T_1(q) = 3*q + 75*q^4 + 969*q^7 + 8964*q^10 + 66975*q^13 + ...

T_2(q) = 9*q^2 + 180*q^5 + 2070*q^8 + 17775*q^11 + 125865*q^14 + ...

then:

3*T_0(q)/T_1(q) = 3*T_1(q)/T_2(q) = T9B(q), the g.f. of A058091:

T9B(q) = 1/q + 5*q^2 - 7*q^5 + 3*q^8 + 15*q^11 - 32*q^14 + 9*q^17 + 58*q^20 + ...

MATHEMATICA

eta[q_] := q^(1/24)*QPochhammer[q]; nmax = 100; R[q_] := (eta[q])^3/ eta[q^3]; a:= CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*3^(IntegerExponent[k, 3] + 1)*q^k/k, {k, 1, nmax}]], {q, 0, 80}], q]; Table[a[[n]], {n, 1, 80}]  (* G. C. Greubel, Jul 03 2018 *)

PROG

(PARI) {a(n)=local(L=sum(m=1, n, 3*sigma(m)*3^valuation(m, 3)*x^m/m)+x*O(x^n)); polcoeff(exp(L), n)}

CROSSREFS

Cf. trisections: A163130 (T_0), A163131 (T_1), A163132 (T_2).

Cf. A058091, A038500, A162584 (variant).

Sequence in context: A138938 A154147 A179545 * A074003 A078844 A144817

Adjacent sequences:  A163126 A163127 A163128 * A163130 A163131 A163132

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 21 2009, Jul 24 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 | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 26 02:56 EDT 2020. Contains 334613 sequences. (Running on oeis4.)