A161293 Number of partitions of n into numbers not divisible by 4 where every part appears at least 2 times.

1, 0, 1, 1, 2, 1, 4, 2, 5, 5, 8, 6, 13, 10, 17, 17, 24, 22, 37, 32, 48, 49, 66, 64, 94, 88, 121, 126, 162, 163, 222, 218, 283, 298, 370, 381, 491, 498, 621, 659, 798, 834, 1035, 1070, 1297, 1384, 1642, 1734, 2093, 2192, 2600, 2785, 3252, 3457, 4085, 4316, 5034, 5406, 6232

Offset

0,5

Comments

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

Number of partitions of n into parts divisible by 2 or 3 except multiples of 8 and 12 are excluded. - Michael Somos, Jul 08 2009

Links

R. H. Hardin, Table of n, a(n) for n=0..802

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

G.f. Product[1+x^(2j)/(1-x^j), j=1..infinity]/Product[1+x^(8j)/(1-x^(4j)), j=1..infinity]. - Emeric Deutsch, Jun 21 2009

Expansion of chi(x^3) * chi(x^6) / (chi(-x^2) * chi(-x^4)) in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Jul 08 2009

Expansion of q^(1/8) * eta(q^6) * eta(q^8) * eta(q^12)/ ( eta(q^2) * eta(q^3) * eta(q^24) ) in powers of q. - Michael Somos, Jul 08 2009

Euler transform of period 24 sequence [ 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, ...]. - Michael Somos, Jul 08 2009

G.f.: Product_{k>0} (1 - x^k + x^(2*k)) * (1 - x^(4*k)) / ( (1 - x^k) * (1 - x^(4*k) + x^(8*k)) ). - Michael Somos, Jul 08 2009

Expansion of psi(x^3) * f(-x^8) / (psi(-x^6) * f(-x^2)) in powers of x where psi(), f() are Ramanujan theta functions. - Michael Somos, Sep 02 2015

Expansion of psi(x^3) * psi(x^4) / (psi(-x^6) * psi(-x^2)) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Sep 02 2015

Example

a(8) = 5 because we have 3311, 22211, 221111, 2(1^6), and 1^8. - Emeric Deutsch, Jun 21 2009
G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 4*x^6 + 2*x^7 + 5*x^8 + 5*x^9 + 8*x^10 + ...
G.f. = 1/q + q^15 + q^23 + 2*q^31 + q^39 + 4*q^47 + 2*q^55 + 5*q^63 + 5*q^71 + ...

Maple

g := (product(1+x^(2*j)/(1-x^j), j = 1 .. 60))/(product(1+x^(8*j)/(1-x^(4*j)), j = 1 .. 60)): gser := series(g, x = 0, 70): seq(coeff(gser, x, n), n = 0 .. 60); # Emeric Deutsch, Jun 21 2009

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3, x^6] QPochhammer[ -x^6, x^12] QPochhammer[ -x^2, x^2] QPochhammer[-x^4, x^4], {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)
a[ n_] := SeriesCoefficient[ (1/2) x^(1/8) EllipticTheta[ 2, 0, x^(3/2)] EllipticTheta[ 2, 0, x^2] / (EllipticTheta[ 2, Pi/4, x] EllipticTheta[ 2, Pi/4, x^3]), {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A) * eta(x^8 + A) * eta(x^12 + A)/ ( eta(x^2 + A) * eta(x^3 + A) * eta(x^24 + A) ), n))}; /* Michael Somos, Jul 08 2009 */

Crossrefs

Sequence in context: A161026 A161077 A339220 * A328163 A217916 A057923

Adjacent sequences: A161290 A161291 A161292 * A161294 A161295 A161296

Keyword

nonn,changed

Author

R. H. Hardin, Jun 06 2009

Extensions

a(0) = 1 added by N. J. A. Sloane, Sep 13 2009

Status

approved