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A053762 Number of 3-colored generalized Frobenius partitions of n. 6
1, 9, 27, 82, 207, 486, 1055, 2205, 4374, 8427, 15696, 28539, 50630, 88119, 150417, 252727, 418068, 682344, 1099343, 1750968, 2758185, 4301682, 6645150, 10175625, 15451744, 23281686, 34819227, 51712860, 76292784, 111850740, 162997314 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

G. E. Andrews, Generalized Frobenius Partitions, AMS Memoir 301, 1984 (sequence is denoted c\phi_3(n)).

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q^(1/8) * (eta(q)^3 + 9 * eta(q^9)^3) / (eta(q)^3 * eta(q^3)) in powers of q. - Michael Somos, Mar 09 2011

Expansion of a(x) / f(-x)^3 in powers of x where a() is a cubic AGM theta function and f() is a Ramanujan theta function. - Michael Somos, Aug 21 2012

Convolution of A000716 and A004016. - Michael Somos, Mar 09 2011

a(n) ~ exp(sqrt(2*n)*Pi)/(4*sqrt(3)*n). - Vaclav Kotesovec, Nov 13 2016

EXAMPLE

1 + 9*x + 27*x^2 + 82*x^3 + 207*x^4 + 486*x^5 + 1055*x^6 + 2205*x^7 + ...

1/q + 9*q^7 + 27*q^15 + 82*q^23 + 207*q^31 + 486*q^39 + 1055*q^47 + 2205*q^55 + ...

MATHEMATICA

nmax = 30; CoefficientList[Series[(Product[(1 - x^k)^3, {k, 1, nmax}] + 9*x*Product[(1 - x^(9*k))^3, {k, 1, nmax}]) / Product[((1 - x^k)^3*(1 - x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 13 2016 *)

a[n_]:= SeriesCoefficient[q^(1/8)*(eta[q]^3 + 9*eta[q^9]^3)/(eta[q]^3* eta[q^3]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Feb 08 2018 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3) / (eta(x + A)^3 * eta(x^3 + A)), n))} /* Michael Somos, Mar 09 2011 */

CROSSREFS

Cf. A000716, A004016, A051136.

Sequence in context: A215690 A255623 A036317 * A126322 A020279 A339725

Adjacent sequences:  A053759 A053760 A053761 * A053763 A053764 A053765

KEYWORD

easy,nonn

AUTHOR

James A. Sellers, Apr 05 2000

STATUS

approved

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Last modified April 16 18:53 EDT 2021. Contains 343050 sequences. (Running on oeis4.)