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A118246 Number of partitions of n such that even parts occur at most once and odd parts occur at most twice. 1
1, 1, 2, 2, 3, 4, 6, 8, 10, 12, 16, 20, 26, 32, 40, 48, 59, 72, 88, 106, 128, 152, 182, 216, 258, 305, 360, 422, 496, 580, 680, 792, 922, 1068, 1238, 1432, 1656, 1908, 2196, 2520, 2892, 3312, 3792, 4330, 4940, 5624, 6400, 7272, 8258, 9361, 10602, 11988, 13548 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also number of partitions of n with no even multiples of 2 and no odd multiples of 3 (i.e. parts equal to 1 or 5 mod 6 and to 2 mod 4). Example: a(7)=8 because we have [7], [6,1], [5,2], [5,1,1], [2,2,2,1], [2,2,1,1,1], [2,1,1,1,1,1] and [1,1,1,1,1,1,1].

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

LINKS

Reinhard Zumkeller and Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 129 terms from Reinhard Zumkeller)

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

G.f.: product((1+x^(2j-1)+x^(4j-2))(1+x^(2j)), j=1..infinity).

G.f.: product([(1-x^(6j-3))(1-x^(4j))]/[(1-x^(2j-1))(1-x^(2j))], j=1..infinity).

G.f.: 1/product((1-x^(1+6j))(1-x^(5+6j))(1-x^(2+4j)), j=0..infinity).

G.f.: product((1+x^j)*(1+x^(2j))/(1+x^(3j)), j=1..infinity). - Vladeta Jovovic, Jul 24 2009

Expansion of chi(-q^3) / (chi(-q) * chi(-q^2)) in powers of q where chi() is a Ramanujan theta function. - Michael Somos, Jun 08 2012

Expansion of eta(q^3) * eta(q^4) / (eta(q) * eta(q^6)) in powers of q. - Michael Somos, Jun 08 2012

Euler transform of period 12 sequence [ 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, ...]. - Michael Somos, Jun 08 2012

a(n) ~ 7^(1/4) * exp(sqrt(7*n/2)*Pi/3) / (2^(9/4)*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Mar 07 2016

EXAMPLE

a(7)=8 because we have [7], [6,1], [5,2], [5,1,1], [4,3], [4,2,1], [3,3,1] and [3,2,1,1].

1 + q + 2*q^2 + 2*q^3 + 3*q^4 + 4*q^5 + 6*q^6 + 8*q^7 + 10*q^8 + 12*q^9 + ...

MAPLE

g:=product((1+x^(2*j-1)+x^(4*j-2))*(1+x^(2*j)), j=1..50): gser:=series(g, x=0, 65): seq(coeff(gser, x, n), n=0..60);

# second Maple program:

a:= proc(n) option remember; `if`(n=0, 1, add(add(

      [0, d$2, 0$2, d$3, 0$2, d$2, 0][1+irem(d, 12)],

       d=numtheory[divisors](j))*a(n-j), j=1..n)/n)

    end:

seq(a(n), n=0..100);  # Alois P. Heinz, Nov 30 2015

MATHEMATICA

QP = QPochhammer; s = QP[q^3]*(QP[q^4]/(QP[q]*QP[q^6])) + O[q]^60; CoefficientList[s, q] (* Jean-Fran├žois Alcover, Nov 30 2015, adapted from PARI *)

nmax = 60; CoefficientList[Series[Product[(1+x^k)*(1+x^(2*k))/(1+x^(3*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^4 + A) / (eta(x + A) * eta(x^6 + A)), n))} /* Michael Somos, Jun 08 2012 */

CROSSREFS

Sequence in context: A266900 A114541 A077114 * A116902 A066447 A035542

Adjacent sequences:  A118243 A118244 A118245 * A118247 A118248 A118249

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Apr 18 2006

STATUS

approved

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Last modified September 28 03:11 EDT 2016. Contains 276599 sequences.