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A104502
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Number of partitions where no part is a multiple of 9.
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18
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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
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OFFSET
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0,3
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COMMENTS
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Coefficients of the B-Dyson Mod 27 identity.
Also partitions where parts are repeated at most 8 times. - Joerg Arndt, Dec 31 2012
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REFERENCES
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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).
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LINKS
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FORMULA
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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))))
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
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
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EXAMPLE
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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 + ...
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MAPLE
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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
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MATHEMATICA
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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;
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 9], 0, 2] ], {n, 0, 46}] (* Robert Price, Jul 29 2020 *)
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PROG
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(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))}
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CROSSREFS
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Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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