A051136 Number of 2-colored generalized Frobenius partitions.

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

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.

Michael 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-François 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