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A261776 Expansion of Product_{k>=1} (1 - x^(10*k))/(1 - x^k). 15

%I

%S 1,1,2,3,5,7,11,15,22,30,41,55,75,98,130,169,220,282,363,460,584,735,

%T 923,1151,1435,1775,2194,2698,3311,4045,4935,5994,7270,8787,10600,

%U 12749,15310,18330,21912,26130,31107,36949,43823,51863,61290,72293,85145,100107

%N Expansion of Product_{k>=1} (1 - x^(10*k))/(1 - x^k).

%C General asymptotic formula (Hagis, 1971): If s > 1 and g.f. = Product_{k>=1} (1 - x^(s*k))/(1 - x^k), then a(n) ~ exp(Pi*sqrt(2*n*(s-1)/(3*s))) * (s-1)^(1/4) / (2 * 6^(1/4) * s^(3/4) * n^(3/4)) * (1 + ((s-1)^(3/2)*Pi/(24*sqrt(6*s)) - 3*sqrt(6*s) / (16*Pi * sqrt(s-1))) / sqrt(n) + ((s-1)^3*Pi^2/(6912*s) - 45*s/(256*(s-1)*Pi^2) - 5*(s-1)/128) / n), minor asymptotic terms added by _Vaclav Kotesovec_, Jan 13 2017

%C The formula in the article by Noureddine Chair: "The Euler-Riemann Gases, and Partition Identities", p. 32, is incorrect (must be s -> s-1 and 24 -> 24*n).

%C Number of partitions in which no part occurs more than 9 times. - _Ilya Gutkovskiy_, May 31 2017

%H Seiichi Manyama, <a href="/A261776/b261776.txt">Table of n, a(n) for n = 0..10000</a>

%H Noureddine Chair, <a href="http://arxiv.org/abs/1306.5415">The Euler-Riemann Gases, and Partition Identities</a>, arXiv:1306.5415 [math-ph], 2013, p. 32.

%H Peter Hagis jr., <a href="https://doi.org/10.1090/S0002-9947-1971-0272735-1">Partitions with a restriction on the multiplicity of the summands</a>, Transactions of the American Mathematical Society, Volume 155, Number 2, April 1971.

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 15.

%F a(n) ~ 3*Pi * BesselI(1, sqrt((24*n + 9)/10) * Pi/2) / (5*sqrt(24*n + 9)) ~ exp(Pi*sqrt(3*n/5)) * 3^(1/4) / (4 * 5^(3/4) * n^(3/4)) * (1 + (3^(3/2)*Pi/(16*sqrt(5)) - sqrt(15)/(8*Pi)) / sqrt(n) + (27*Pi^2/2560 - 25/(128*Pi^2) - 45/128) / n). - _Vaclav Kotesovec_, Aug 31 2015, extended Jan 14 2017

%F a(n) = (1/n)*Sum_{k=1..n} A284344(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 25 2017

%t nmax = 50; CoefficientList[Series[Product[(1 - x^(10*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

%t Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 10], 0, 2] ], {n, 0, 47}] (* _Robert Price_, Jul 29 2020 *)

%o (PARI) Vec(prod(k=1, 51, (1 - x^(10*k))/(1 - x^k)) + O(x^51)) \\ _Indranil Ghosh_, Mar 25 2017

%Y Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

%Y Cf. A261772, A145707, A320612.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Aug 31 2015

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Last modified July 28 14:19 EDT 2021. Contains 346335 sequences. (Running on oeis4.)