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 A219601 Number of partitions of n in which no parts are multiples of 6. 16
 1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 65, 85, 111, 143, 184, 234, 297, 374, 470, 586, 729, 902, 1113, 1367, 1674, 2042, 2485, 3013, 3645, 4395, 5288, 6344, 7595, 9070, 10809, 12852, 15252, 18062, 21352, 25191, 29671, 34884, 40948, 47985, 56146, 65592 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Also partitions where parts are repeated at most 5 times. [Joerg Arndt, Dec 31 2012] LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Arkadiusz Wesolowski) 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. Eric Weisstein's World of Mathematics, Partition Function b_k FORMULA G.f.: P(x^6)/P(x), where P(x) = prod(k>=1, 1-x^k). a(n) ~ Pi*sqrt(5) * BesselI(1, sqrt(5*(24*n + 5)/6) * Pi/6) / (3*sqrt(24*n + 5)) ~ exp(Pi*sqrt(5*n)/3) * 5^(1/4) / (12 * n^(3/4)) * (1 + (5^(3/2)*Pi/144 - 9/(8*Pi*sqrt(5))) / sqrt(n) + (125*Pi^2/41472 - 27/(128*Pi^2) - 25/128) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017 a(n) = (1/n)*Sum_{k=1..n} A284326(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017 EXAMPLE 7 = 7   = 5 + 2   = 5 + 1 + 1   = 4 + 3   = 4 + 2 + 1   = 4 + 1 + 1 + 1   = 3 + 3 + 1   = 3 + 2 + 2   = 3 + 2 + 1 + 1   = 3 + 1 + 1 + 1 + 1   = 2 + 2 + 2 + 1   = 2 + 2 + 1 + 1 + 1   = 2 + 1 + 1 + 1 + 1 + 1   = 1 + 1 + 1 + 1 + 1 + 1 + 1 so a(7) = 14. MATHEMATICA m = 47; f[x_] := (x^6 - 1)/(x - 1); g[x_] := Product[f[x^k], {k, 1, m}]; CoefficientList[Series[g[x], {x, 0, m}], x] (* Arkadiusz Wesolowski, Nov 27 2012 *) Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 6], 0, 2] ], {n, 0, 47}] (* Robert Price, Jul 28 2020 *) PROG (PARI) for(n=0, 47, A=x*O(x^n); print1(polcoeff(eta(x^6+A)/eta(x+A), n), ", ")) CROSSREFS Cf. A097797. Cf. A261770, A261736, A320608. Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546. Sequence in context: A326977 A035967 A097797 * A035975 A035984 A035994 Adjacent sequences:  A219598 A219599 A219600 * A219602 A219603 A219604 KEYWORD easy,nonn AUTHOR Arkadiusz Wesolowski, Nov 23 2012 STATUS approved

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Last modified August 12 06:22 EDT 2020. Contains 336438 sequences. (Running on oeis4.)