A104502 Number of partitions where no part is a multiple of 9.

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 54, 74, 96, 128, 165, 216, 275, 354, 447, 569, 712, 896, 1113, 1388, 1712, 2117, 2595, 3186, 3882, 4735, 5739, 6959, 8392, 10121, 12150, 14582, 17429, 20823, 24789, 29494, 34979, 41456, 48993, 57856, 68148, 80204

Offset: 0

Author

Eric W. Weisstein, Mar 11 2005

Keywords

• nonn

Comments

Coefficients of the B-Dyson Mod 27 identity.

Also partitions where parts are repeated at most 8 times. - Joerg Arndt, Dec 31 2012

Examples

			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 + ...
B(q) = q + q^4 + 2*q^7 + 3*q^10 + 5*q^13 + 7*q^16 + 11*q^19 + 15*q^22 + ...
		

References

• 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).

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.
• Eric Weisstein's World of Mathematics, Dyson Mod 27 Identities

Crossrefs

Cf. A062246, A104501, A104503, A104504.

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

Cf. A112193, A261733, A320611.

Programs

Maple

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

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 - x^(9*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
a[n_] := a[n] = (1/n) Sum[DivisorSum[k, Boole[!Divisible[#, 9]] #&] a[n-k], {k, 1, n}]; a[0] = 1;
a /@ Range[0, 50] (* Jean-François Alcover, Oct 01 2019, after Seiichi Manyama *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 9], 0, 2] ], {n, 0, 46}] (* Robert Price, Jul 29 2020 *)

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 */

PARI

{A116607(n)=sigma(n)-if(n%9==0, 9*sigma(n/9))}
{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 */

Formula

Expansion of q^(-1/3) * eta(q^9) / eta(q) in powers of q. - Michael Somos, Jan 09 2006

Euler transform of period 9 sequence [1, 1, 1, 1, 1, 1, 1, 1, 0, ...]. - Michael Somos, Jan 09 2006

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

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))))

G.f. A(x) = 1/g.f. A062246.

Logarithmic derivative yields A116607 (sum of the divisors of n which are not divisible by 9). - Paul D. Hanna, Jun 13 2011

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

a(n) = (1/n)*Sum_{k=1..n} A116607(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

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

Extensions

Simplified definition. - N. J. A. Sloane, Oct 20 2019