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 A035985 Number of partitions of n into parts not a multiple of 7. Also number of partitions with at most 6 parts of size 1 and differences between parts at distance 9 are greater than 1. 16
 1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, 51, 70, 90, 119, 153, 199, 252, 324, 406, 515, 642, 804, 994, 1236, 1517, 1869, 2282, 2791, 3387, 4118, 4970, 6006, 7217, 8673, 10374, 12411, 14780, 17601, 20883, 24766, 29274, 34588, 40741, 47964, 56319, 66080, 77350 (list; graph; refs; listen; history; text; internal format)
 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 Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546. Cf. A320609. Sequence in context: A288255 A325853 A035976 * A035995 A036006 A027341 Adjacent sequences:  A035982 A035983 A035984 * A035986 A035987 A035988 KEYWORD nonn,easy AUTHOR EXTENSIONS Definition simplified by N. J. A. Sloane, Oct 20 2019 STATUS approved

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Last modified June 14 14:50 EDT 2021. Contains 345025 sequences. (Running on oeis4.)