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A051136 Number of 2-colored generalized Frobenius partitions. 6
1, 4, 9, 20, 42, 80, 147, 260, 445, 744, 1215, 1944, 3059, 4740, 7239, 10920, 16286, 24028, 35110, 50844, 73010, 104028, 147144, 206700, 288501, 400232, 552037, 757288, 1033495, 1403508, 1897088, 2552812, 3420527, 4564500, 6067265 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

REFERENCES

G. E. Andrews, "Generalized Frobenius Partitions," AMS Memoir 301, 1984 (sequence is denoted c\phi_2(n)).

G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 67, Eq. (7.20). MR0858826 (88b:11063)

LINKS

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

Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of phi(q) / f(-q)^2 in powers of q where phi(), f() are Ramanujan theta functions.

Expansion of q^(1/12) * eta(q^2)^5 / (eta(q)^4 * eta(q^4)^2) in powers of q. - Michael Somos, Apr 25 2003

Euler transform of period 4 sequence [4, -1, 4, 1, ...]. - Michael Somos, Apr 25 2003

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 24^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g(t) is g.f. for A137828.

G.f.: Product_{k>0} (1 -x^(4*k-2)) / ((1 - x^(2*k-1))^4 * (1 - x^(4*k))). [Andrews, Memoir, p. 13, equation (5.17)]

G.f.: Product_{k>0} (1 + x^k)^3 / ((1 - x^k) * (1 + x^(2*k))^2). - Michael Somos, Feb 12 2008

a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*sqrt(3)*n). - Vaclav Kotesovec, Aug 31 2015

EXAMPLE

1 + 4*x + 9*x^2 + 20*x^3 + 42*x^4 + 80*x^5 + 147*x^6 + 260*x^7 + ...

1/q + 4*q^11 + 9*q^23 + 20*q^35 + 42*q^47 + 80*q^59 + 147*q^71 + ...

MATHEMATICA

nmax = 100; CoefficientList[Series[Product[(1 + x^(2*k-1)) / ((1 - x^(2*k-1))^3 * (1 - x^(4*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)

QP = QPochhammer; s = QP[q^2]^5 / QP[q]^4 / QP[q^4]^2 + O[q]^40; CoefficientList[s, q] (* Jean-Fran├žois Alcover, Nov 09 2015, adapted from PARI *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / eta(x + A)^4 / eta(x^4 + A)^2, n))} /* Michael Somos, Feb 12 2008 */

CROSSREFS

Cf. A000041, A053762, A247663, A247664.

Sequence in context: A084639 A272268 A284736 * A156321 A276778 A133095

Adjacent sequences:  A051133 A051134 A051135 * A051137 A051138 A051139

KEYWORD

easy,nonn

AUTHOR

James A. Sellers

STATUS

approved

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Last modified January 25 01:39 EST 2020. Contains 331229 sequences. (Running on oeis4.)