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%I #92 Jan 23 2023 13:13:48
%S 1,1,2,3,5,6,10,13,19,25,34,44,60,76,100,127,164,205,262,325,409,505,
%T 628,769,950,1156,1414,1713,2081,2505,3026,3625,4352,5192,6200,7364,
%U 8756,10357,12258,14450,17034,20006,23500,27510,32200,37582,43846,51022
%N Number of partitions of n in which no parts are multiples of 5.
%C Also number of partitions with at most 4 parts of size 1 and differences between parts at distance 6 are greater than 1.
%C Also number of partitions of n where no part appears more than four times.
%C Case k=7, i=5 of Gordon Theorem.
%D G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.
%H Seiichi Manyama, <a href="/A035959/b035959.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H Riccardo Aragona, Roberto Civino, and Norberto Gavioli, <a href="https://arxiv.org/abs/2301.06347">A modular idealizer chain and unrefinability of partitions with repeated parts</a>, arXiv:2301.06347 [math.RA], 2023.
%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.
%H G. N. Watson, <a href="http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002174499">Ramanujans Vermutung ueber Zerfaellungsanzahlen</a>, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. See the expression Y = C/B in the notation of p. 106. [Added by _N. J. A. Sloane_, Nov 13 2009]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PartitionFunctionb.html">Partition Function b_k.</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Glaisher%27s_theorem">Glaisher's Theorem</a>
%F G.f.: Product_{j>=1} (1 + x^j + x^2j + x^3j + x^4j). - _Jon Perry_, Mar 30 2004
%F G.f.: Product_{n>0, n==1, 2, 3, 4 mod 5} 1/(1-q^n).
%F Given g.f. A(x) then B(x) = x * A(x^3)^2 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^3 + v^3 - u*v - 5*u^2*v^2. - _Michael Somos_, May 28 2006
%F Given g.f. A(x) then B(x) = x * A(x^3)^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v + 5*v^2*(u + w) - (u^2 + u*w + w^2). - _Michael Somos_, May 28 2006
%F Euler transform of period 5 sequence [ 1, 1, 1, 1, 0, ...]. - _Michael Somos_, May 28 2006
%F G.f.: Product_{k > 0} P5(x^k) where P5 is 5th cyclotomic polynomial.
%F Convolution inverse is A145466. - _Michael Somos_, Jun 26 2014
%F a(n) ~ 2*Pi * BesselI(1, 2*sqrt((6*n + 1)/5) * Pi/3) / (5*sqrt(6*n + 1)) ~ exp(2*Pi*sqrt(2*n/15)) / (3^(1/4) * 10^(3/4) * n^(3/4)) * (1 + (Pi/(3*sqrt(15)) - 3*sqrt(15)/(16*Pi)) / sqrt(2*n) + (Pi^2/540 - 225/(1024*Pi^2) - 5/32) / n). - _Vaclav Kotesovec_, Aug 31 2015, extended Jan 14 2017
%F a(n) = (1/n)*Sum_{k=1..n} A116073(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 25 2017
%F G.f.: exp(Sum_{k>=1} x^k*(1 + x^k + x^(2*k) + x^(3*k))/(k*(1 - x^(5*k)))). - _Ilya Gutkovskiy_, Aug 15 2018
%e G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 13*x^7 + 19*x^8 + ...
%e G.f. = q + q^7 + 2*q^13 + 3*q^19 + 5*q^25 + 6*q^31 + 10*q^37 + 13*q^43 + ...
%e a(6) counts these partitions: 6, 42, 411, 33, 321, 3111, 2211, 21111, 111111. - _Clark Kimberling_, Mar 09 2014
%t max = 47; f[x_] := (x^5-1)/(x-1); g[x_] := Product[f[x^k], {k, 1, max}]; CoefficientList[ Series[g[x], {x, 0, max}], x] (* _Jean-François Alcover_, Nov 29 2011, after _Michael Somos_ *)
%t t = Flatten[Table[5 n + r, {n, 0, 60}, {r, 1, 4}]]; p[n_] := IntegerPartitions[n, All, t]; Table[p[n], {n, 0, 8}] (* shows partitions *)
%t a[n_] := Length@p@n; a /@ Range[0, 50] (* _Clark Kimberling_, Mar 09 2014 *)
%t nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 31 2015 *)
%t QP = QPochhammer; s = QP[q^5]/QP[q] + O[q]^50; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 25 2015, after _Michael Somos_ *)
%t Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 5], 0, 2] ], {n, 0, 47}] (* _Robert Price_, Jul 28 2020 *)
%t Table[Count[IntegerPartitions[n],_?(NoneTrue[Mod[#,5]==0&])],{n,0,50}] (* _Harvey P. Dale_, Dec 25 2021 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( eta(x^5 + x * O(x^n)) / eta(x + x * O(x^n)), n))}; /* _Michael Somos_, May 28 2006 */
%o (Haskell)
%o a035959 = p a047201_list where
%o p _ 0 = 1
%o p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%o -- _Reinhard Zumkeller_, Dec 17 2011
%Y Cf. A061198, A061199, A047201.
%Y Cf. A000009 (m=2), A000726 (m=3), A001935 (m=4), A219601 (m=6), A035985 (m=7), A261775 (m=8), A104502 (m=9), A261776 (m=10).
%Y Cf. A096938, A133563, A320607.
%Y Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.
%K nonn,easy,nice
%O 0,3
%A _Olivier Gérard_
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