OFFSET
6,3
REFERENCES
Newman, Morris; Construction and application of a class of modular functions. II. Proc. London Math. Soc. (3) 9 1959 373-387.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 6..1000
Morris Newman, Construction and application of a class of modular functions, II, Proc. London Math. Soc. (3) 9 1959 373-387. [Annotated scanned copy, barely legible]
FORMULA
Expansion of eta(q^15)^13 / (eta(q) * eta(q^3)^5 * eta(q^5)^7) in powers of q.
Expansion of (c(q^5)^2 / (3 * c(q)))^2 / (b(q) * b(q^5)) in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Jun 10 2012
Euler transform of period 15 sequence [1, 1, 6, 1, 8, 6, 1, 1, 6, 8, 1, 6, 1, 1, 0, ...]. - Michael Somos, Nov 10 2005
a(n) ~ exp(4*Pi*sqrt(2*n/15)) / (2^(1/4) * 3^(17/4) * 5^(13/4) * n^(3/4)). - Vaclav Kotesovec, Apr 09 2018
EXAMPLE
q^6 + q^7 + 2*q^8 + 8*q^9 + 10*q^10 + 24*q^11 + 53*q^12 + 74*q^13 + 153*q^14 + ...
MAPLE
with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: aa:=etr(n-> [1, 1, 6, 1, 8, 6, 1, 1, 6, 8, 1, 6, 1, 1, 0] [modp(n-1, 15)+1]): a:=n-> aa(n-6): seq(a(n), n=6..42); # Alois P. Heinz, Sep 08 2008
MATHEMATICA
etr[p_] := Module[{b}, b[n_] := b[n] = Module[{d, j}, If[n == 0, 1, Sum [Sum [d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]]; b]; aa = etr[ Function[n, {1, 1, 6, 1, 8, 6, 1, 1, 6, 8, 1, 6, 1, 1, 0}[[Mod[n-1, 15] + 1]]]]; a[n_] := aa[n-6]; Table[a[n], {n, 6, 41}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)
PROG
(PARI) {a(n) = local(A); if( n<6, 0, n -= 6; A = x * O(x^n); polcoeff( eta(x^15 + A)^13 / (eta(x + A) * eta(x^3 + A)^5 * eta(x^5 + A)^7), n))} /* Michael Somos, Nov 10 2005 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jan 14 2001
STATUS
approved