A131945 Number of partitions of n where odd parts are distinct or repeated once.

1, 1, 2, 2, 4, 5, 8, 10, 15, 18, 26, 32, 45, 55, 74, 90, 119, 145, 188, 228, 291, 351, 442, 532, 664, 796, 982, 1172, 1435, 1708, 2076, 2462, 2972, 3512, 4214, 4966, 5929, 6965, 8272, 9688, 11457, 13383, 15762, 18362, 21543, 25031, 29264, 33922, 39533, 45717

Offset

0,3

Comments

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

Also number of partitions of n such that every part is not congruent to 3 mod 6. More generally, g.f. for number of partitions of n such that every odd part occurs at most m times is product_{n=1..inf} (1-q^((m+1)*(2*n-1)))/(1-q^n). Similarly, g.f. for number of partitions of n such that every even part occurs at most m times is product_{n=1..inf} (1-q^((2*m+2)*n))/(1-q^n). - Vladeta Jovovic, Aug 01 2007

Links

Brian Drake and Seiichi Manyama, Table of n, a(n) for n = 0..1000 (first 101 terms from Brian Drake)

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

G.f.: product_{n=1..inf} (1-q^(6n-3))/(1-q^n).

Expansion of chi(-x^3) / f(-x) in powers of x where chi(), f() are Ramanujan theta functions. - Michael Somos, Aug 05 2007

Expansion of q^(1/6) * eta(q^3) / (eta(q) * eta(q^6)) in powers of q. - Michael Somos, Aug 05 2007

Euler transform of period 6 sequence [ 1, 1, 0, 1, 1, 1, ...]. - Michael Somos, Aug 05 2007

a(n) ~ sqrt(5) * exp(sqrt(5*n)*Pi/3) / (12*n). - Vaclav Kotesovec, Dec 11 2016

Example

a(6) = 8 because we have 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 2+2+2 and 2+2+1+1. The three excluded partitions of 6 are 3+1+1+1, 2+1+1+1+1 and 1+1+1+1+1+1.
G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 5*x^5 + 8*x^6 + 10*x^7 + 15*x^8 + ...
G.f. = 1/q + q^5 + 2*q^11 + 2*q^17 + 4*q^23 + 5*q^29 + 8*q^35 + 10*q^41 + ...

Maple

A:= series(product( (1-q^(6*n-3))/(1-q^n), n=1..20), q, 21): seq(coeff(A, q, i), i=0..20);

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] / (QPochhammer[ x] QPochhammer[ x^6]), {x, 0, n}]; (* Michael Somos, Jan 29 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^6] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 11 2015 *)
nmax = 50; CoefficientList[Series[Product[1 / ((1-x^k) * (1+x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 11 2016 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)), n))}; /* Michael Somos, Aug 05 2007 */

Crossrefs

Cf. A006950, A131942, A279320.

Sequence in context: A326446 A323530 A271593 * A240308 A326526 A035951

Adjacent sequences: A131942 A131943 A131944 * A131946 A131947 A131948

Keyword

easy,nonn

Author

Brian Drake, Jul 30 2007

Status

approved