OFFSET
0,3
COMMENTS
Case k=10, i=7 of Gordon Theorem.
REFERENCES
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 15.
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. This sequence arises as the coefficients of Y = C/B on p. 118.
FORMULA
Euler transform of period 7 sequence [1, 1, 1, 1, 1, 1, 0, ...]. - Michael Somos, Jan 17 2006
Given g.f. A(x), then B(x)=x*A(x^4) satisfies 0=f(B(x), B(x^3)) where f(u, v)=(u^4+v^4)-u*v*(1+3*u*v+7*(u*v)^2).
G.f.: Product_{k>0} (1-x^(7k))/(1-x^k).
Given g.f. A(x) then B(x)=x*A(x)^4 satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u,v,w)= (u^2+u*w+w^2) -v -8*v*(u+v+w) -49*v^2*(u+w). - Michael Somos, May 28 2006
G.f. is product k>0 P7(x^k) where P7 is 7th cyclotomic polynomial.
Expansion of q^(-1/4)eta(q^7)/eta(q) in powers of q. - Michael Somos, Jan 17 2006
a(n) ~ 2*Pi * BesselI(1, sqrt((4*n + 1)/7) * Pi) / (7*sqrt(4*n + 1)) ~ exp(2*Pi*sqrt(n/7)) / (2 * 7^(3/4) * n^(3/4)) * (1 + (Pi/(4*sqrt(7)) - 3*sqrt(7)/(16*Pi)) / sqrt(n) + (Pi^2/224 - 105/(512*Pi^2) - 15/64) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A113957(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
EXAMPLE
B(x) = x +x^5 +2*x^9 +3*x^13 +5*x^17 +7*x^21 +11*x^25 +14*x^29 +...
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1 - x^(7*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
QP = QPochhammer; s = QP[q^7]/QP[q] + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 7], 0, 2] ], {n, 0, 47}] (* Robert Price, Jul 28 2020 *)
PROG
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^7+A)/eta(x+A), n))} /* Michael Somos, Jan 17 2006 */
(PARI) Vec(prod(k=1, 50, (1 - x^(7*k))/(1 - x^k)) + O(x^51)) \\ Indranil Ghosh, Mar 25 2017
(PARI) A035985_upto(N, q='x+O('x^N))=Vec(eta(q^7)/eta(q)) \\ M. F. Hasler, Dec 09 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Definition simplified by N. J. A. Sloane, Oct 20 2019
STATUS
approved