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%I #59 Jul 29 2020 20:05:59
%S 1,1,2,3,5,7,11,15,22,29,41,54,74,96,128,165,216,275,354,447,569,712,
%T 896,1113,1388,1712,2117,2595,3186,3882,4735,5739,6959,8392,10121,
%U 12150,14582,17429,20823,24789,29494,34979,41456,48993,57856,68148,80204
%N Number of partitions where no part is a multiple of 9.
%C Coefficients of the B-Dyson Mod 27 identity.
%C Also partitions where parts are repeated at most 8 times. - _Joerg Arndt_, Dec 31 2012
%D F. J. Dyson, A walk through Ramanujan's garden, pp. 7-28 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988, see p. 15, eq. (11).
%H Seiichi Manyama, <a href="/A104502/b104502.txt">Table of n, a(n) for n = 0..10000</a>
%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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DysonMod27Identities.html">Dyson Mod 27 Identities</a>
%F Expansion of q^(-1/3) * eta(q^9) / eta(q) in powers of q. - _Michael Somos_, Jan 09 2006
%F Euler transform of period 9 sequence [1, 1, 1, 1, 1, 1, 1, 1, 0, ...]. - _Michael Somos_, Jan 09 2006
%F Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^3 + v^3 - u*v - 3*(u*v)^2. - _Michael Somos_, Jan 09 2006
%F G.f.: Product_{k>0} (1-x^(9k))/(1-x^k) = 1 + 1/(1-x)*(Sum_{k>0} x^(k^2+k) Product_{i=1..k} (1+x^i+x^(2i))/((1-x^(2i))*(1-x^(2i+1))))
%F G.f. A(x) = 1/g.f. A062246.
%F Logarithmic derivative yields A116607 (sum of the divisors of n which are not divisible by 9). - _Paul D. Hanna_, Jun 13 2011
%F a(n) ~ 2*Pi * BesselI(1, 4*sqrt(3*n + 1) * Pi/9) / (9*sqrt(3*n + 1)) ~ exp(4*Pi*sqrt(n/3)/3) / (sqrt(2) * 3^(7/4) * n^(3/4)) * (1 + (2*Pi/(9*sqrt(3)) - 9*sqrt(3)/(32*Pi)) / sqrt(n) + (2*Pi^2/243 - 405/(2048*Pi^2) - 5/16) / n). - _Vaclav Kotesovec_, Aug 31 2015, extended Jan 14 2017
%F a(n) = (1/n)*Sum_{k=1..n} A116607(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 25 2017
%F G.f. is a period 1 Fourier series that satisfies f(-1 / (81 t)) = 1/3 g(t) where g() is the g.f. for A062246. - _Michael Somos_, Jun 27 2017
%e G.f. = 1 + q + 2*q^2 + 3*q^3 + 5*q^4 + 7*q^5 + 11*q^6 + 15*q^7 + 22*q^8 + 29*q^9 + ...
%e B(q) = q + q^4 + 2*q^7 + 3*q^10 + 5*q^13 + 7*q^16 + 11*q^19 + 15*q^22 + ...
%p seq(coeff(series(mul((1-x^(9*k))/(1-x^k),k=1..n),x,n+1), x, n), n = 0 .. 50); # _Muniru A Asiru_, Sep 29 2018
%t nmax = 50; CoefficientList[Series[Product[(1 - x^(9*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 31 2015 *)
%t a[n_] := a[n] = (1/n) Sum[DivisorSum[k, Boole[!Divisible[#, 9]] #&] a[n-k], {k, 1, n}]; a[0] = 1;
%t a /@ Range[0, 50] (* _Jean-François Alcover_, Oct 01 2019, after _Seiichi Manyama_ *)
%t Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 9], 0, 2] ], {n, 0, 46}] (* _Robert Price_, Jul 29 2020 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^9 + A) / eta(x + A), n))}; /* _Michael Somos_, Jan 09 2006 */
%o (PARI) {A116607(n)=sigma(n)-if(n%9==0, 9*sigma(n/9))}
%o {a(n)=polcoeff(exp(sum(k=1, n+1, A116607(k)*x^k/k+x*O(x^n))), n)} /* _Paul D. Hanna_, Jun 13 2011 */
%Y Cf. A062246, A104501, A104503, A104504.
%Y Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.
%Y Cf. A112193, A261733, A320611.
%K nonn
%O 0,3
%A _Eric W. Weisstein_, Mar 11 2005
%E Simplified definition. - _N. J. A. Sloane_, Oct 20 2019
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